
A \(-module\)  is a spectral functor
\(\mathcal {S}p.\) 


A \((\) -bimodule is a spectral functor
\( {op} \wedge \mathcal {S}p,\) 
where \( {op} \wedge  is the spectral category with as objects ordered pairs \) (c,d)\(, where \) c  and \(d \in , and with mapping spectrum\)\(\mathrm {Map}_{{op} \wedge \big ((c_1,d_1),(c_2,d_2)\big ) = \mathrm {Map}_c_1,c_2) \wedge \mathrm {Map}_d_1,d_2).}\)
Definition 2.2 (
[1]}) 

Let \( be a be a pointwise-cofibrant spectral category (i.e., \) c,d)\( is a cofibrant spectrum for all objects \) c,d\() and \)  a \((\) -bimodule that is pointwise-cofibrant as a spectral category. The topological Hochschild homology of \( with coefficients in \) , denoted \(\mathrm {THH}(\) , is the geometric realization of the cyclic bar construction \(N^{cy}_\bullet ( \) , i.e., the simplicial spectrum that at level \(n\)  is
\(N^{cy}_n(  = \bigvee _{c_0,...,c_n} c_0,c_1) \wedge c_1,c_2) \wedge ... \wedge c_n,c_0),\) 
with faces given by composition in \( or the action of \)  on \( (together with a cyclic permutation for the last face) and degeneracies given by the unit,i.e.,\)\(\mathrm {THH}( = |N^{cy}_\bullet ( |.\)\(\)
Ponto's key insight was that this definition could be axiomatized. Concretely, let \(\mathrm {Ho}({M}\mathrm {od})\)  be the bicategory with pointwise-cofibrant small spectral categories as objects and with the morphism category from \( to \)\( equal to the homotopy category of \) ()\(-bimodules, \) Ho(Mod())\(.For every pointwise-cofibrant spectral category \) , topological Hochschild homology gives rise to a collection of functors
\(\mathrm {THH}: \mathrm {Ho}({M}\mathrm {od})( = \mathrm {Ho}({M}\mathrm {od}_{(}) \rightarrow \mathrm {Ho}(\mathrm {Sp}): \mathrm {THH}( ,\) 
satisfying certain properties, which become the axioms in the definition of a shadow, which we recall now.
Definition 2.3 (
[1]}) 

Let \({B}\)  be a bicategory with associator \(a\) , left unit \(l\) , and right unit \(r\) , and let \( be a category.A {\it shadow} on \) B\( with values in \)  is a functor
\(\langle \langle - \rangle \rangle  : \coprod _{X \in \mathrm {Obj}({B})} {B}(X,X) \rightarrow \) 
that satisfies the following conditions.
For every pair of 1-morphisms \(F: X \rightarrow Y\)  and \(G: Y \rightarrow X\)  in \({B}\)  that are composable in either order, there is a natural isomorphism
\(\theta : \langle \langle FG \rangle \rangle  \xrightarrow{} \langle \langle GF \rangle \rangle \) 
such that for all \(F:X \rightarrow Y\) , \(G:Y \rightarrow Z\) , \(H: Z \rightarrow X\) , and \(K:X\rightarrow X\)  the following diagrams in \( commute up to natural isomorphism.\)
\( \begin{tikzcd}[row sep=0.5in, column sep=0.5in]\langle \langle H(GF) \rangle \rangle  [r, "\theta "] [d, "\langle \langle a \rangle \rangle "^{\prime }] & \langle \langle (GF)H \rangle \rangle  [r, "\langle \langle a \rangle \rangle "] & \langle \langle G(FH) \rangle \rangle  \\\langle \langle (HG)F \rangle \rangle  [r, "\theta "] & \langle \langle F(HG) \rangle \rangle  [r, "\langle \langle a \rangle \rangle "] & \langle \langle (FH)G \rangle \rangle  [u, "\theta "^{\prime }]\end{tikzcd}\) 
\( \begin{tikzcd}[row sep=0.5in, column sep=0.5in]\langle \langle KU_X \rangle \rangle  [r, "\theta "] [dr, "\langle \langle r \rangle \rangle "^{\prime }] & \langle \langle U_XK \rangle \rangle  [d, "\langle \langle l \rangle \rangle "] [r, "\theta "] & \langle \langle KU_X \rangle \rangle  [dl, "\langle \langle r \rangle \rangle "] \\& \langle \langle K \rangle \rangle  &\end{tikzcd}\) 


The primary example of a shadow is \(\mathrm {THH}\)  of spectral categories.
Theorem 2.6 (
[1]}) 
Topological Hochschild homology of spectral categories is a shadow, i.e., the family of functors
\(\Big \lbrace  \mathrm {THH}(-): \mathrm {Ho}({M}\mathrm {od}_{(}) \rightarrow \mathrm {Ho}(\mathrm {Sp}): \mathrm {THH}( \mid \mathrm {Ob}\,\mathrm {Ho}({M}\mathrm {od})\Big \rbrace \) 
is such that for all pointwise-cofibrant spectral categories \( and \)\(, \) ( )\(-bimodules \) , and \((, \) -bimodules \(\) , there is a natural isomorphism
\(\theta : \mathrm {THH}( _{} )\xrightarrow{}\mathrm {THH}(, \wedge _{  satisfying the conditions of {Def:shadow}.}\) 
It is common to simplify notation and to write
\(\mathrm {THH}(=\mathrm {THH}(,\) 
where we consider \( as a bimodule over itself in the obvious way. Note that computing THH of a spectral category yields a spectrum, i.e., an object in the enriching category.\) The axioms of a shadow suffice to prove interesting properties that are known to hold for \(\mathrm {THH}\) . For example, Ponto and Campbell prove a general form of Morita equivalence for shadows [1]} that implies the classical Morita equivalence for ring spectra [1]}.
In thm:main theorem we show that these properties stem from a deep connection between shadows and \(\mathrm {THH}\) . However, in order to understand this relation we need a higher categorical approach to shadows that moves away from point-set models. Laying the foundations of such an approach is the goal of the next subsection.
\(\mathrm {THH}\)  of enriched \(\infty \) -categories
In this subsection we review the definition of \(\mathrm {THH}\)  of an enriched \(\infty \) -category.
Before we move on to the enriched setting, it is valuable to explain why we cannot simply generalize Def:thh of spectral cat.
An \(\infty \) -category \( is a simplicial set satisfying additional conditions, the so-called ``inner horn lifting conditions" \cite {lurie2009htt}. These lifting conditions imply that for any two objects (i.e., \) 0\(-simplices) \) x\( and \) y\(, the simplicial set \) Map(x,y)\( given by the fiber of the map \) 1  over the point \((x,y)\)  is a Kan complex and thus a homotopy-meaningful mapping space. There is no direct composition map, however. Instead, for three objects \(x,y,z\) , there is a zig-zag
\(\mathrm {Map}(x,y) \times \mathrm {Map}(y,z) \xleftarrow{} \mathrm {Map}(x,y,z) \rightarrow \mathrm {Map}(x,z)\) 
where \(\mathrm {Map}(x,y,z)\)  is the fiber of the map \({\Delta ^2} \rightarrow 3\)  over \((x,y,z)\) , and the lifting conditions imply that the first map is a trivial Kan fibration.
This observation has two unfortunate implications.

We cannot just define an enriched \(\infty \) -category in terms of the existence of composition maps and thus need a more complicated notion of enrichment.

We cannot just define \(\mathrm {THH}\)  via the cyclic bar construction, as the simplicial structure relies on the existence of a direct composition map.

Fortunately, Gepner-Haugseng [6]} and Hinich [7]} have developed notions of \(\infty \) -categories enriched in a monoidal \(\infty \) -category, both of which are based on an operadic approach to classical enriched category theory, generalized to the \(\infty \) -categorical setting. Their constructions are very powerful and have been used to prove deep results about enriched \(\infty \) -categories, but are also very intricate and can be difficult to use for computations.
As in this article we need a notion of enrichment only to study \(\mathrm {THH}\) , we can focus on the case where the enrichment category is not just monoidal, but actually symmetric monoidal.
There is a much more convenient way of defining enriched \(\infty \) -categories in the symmetric case, due to Berman [8]}This approach can be in fact generalized to encompass the monoidal case, however, we do not require this generality., which also provides a natural framework for a generalization of \(\mathrm {THH}\) .
The key idea is that of a bypass operation, which we define next. Recall that a directed multigraph is a pair of sets \(\Gamma = (S,E)\)  of vertices and of directed edges between them. For two multigraphs \(\Gamma _1, \Gamma _2\)  with the same set of vertices \(S\) , there is a bypass operation from \(\Gamma _1\)  to \(\Gamma _2\)  if \(\Gamma _2\)  can be obtained from \(\Gamma _1\)  by a sequence of the following combinatorial moves.

Adding a loop

Replacing a path of consecutive edges \(e_1, ... , e_n\)  by a single edge \(e\) .

For a precise definition, see [8]}.
Definition 2.7 Let \(S\)  be any set. The bypass category on \(S\) , \({B}\mathrm {ypass}_S\) , has as objects all directed multigraphs with fixed vertex set \(S\)  and a finite set of edges and as morphisms all bypass operations between such.

Note that \({B}\mathrm {ypass}_S\)  admits a symmetric monoidal product \(\otimes \) , where the set of edges of \(\Gamma _1 \otimes \Gamma _2\)  is the disjoint union of the sets of edges of \(\Gamma _1\)  and \(\Gamma _2\) , and the unit is the multigraph with no edges.
Definition 2.8 
Let \( be a symmetric monoidal \)\(-category. A {\it -enriched \infty -category with set of objects S} is a symmetric monoidal functor of \)\(-categories\)\( {B}\mathrm {ypass}_S \rightarrow \)\(\)
The bypass category is designed to encode the structure of an enriched category. For example, if \((X_0,...,X_n)\)  denotes the multigraph with \(n+1\)  vertices \(X_0,..., X_n\)  and a unique edge from \(X_i\)  to \(X_{i+1}\)  for every \(i\) , then
\(X_0, ..., X_n) \in \) 
can be viewed as the object in \( of sequences of \) n\( composable maps\) X0 ... Xn\(. In particular \) X0,X1)\( plays the role of the mapping object. We can then use a bypass operation of type (2) above to get the desired composition map\)\(X_0,X_1,X_2) \rightarrow X_0,X_2).\)\(The following result should thus not be very surprising.\)
Proposition 2.9 ([8]}) 
If \( is a symmetric monoidal \)\(-category, then the definition of \) -enriched \(\infty \) -category given here agrees with that in [6]}.

We can use this simplified definition of an enriched \(\infty \) -category (and in particular the fact that \({B}\mathrm {ypass}_S\)  is a category) to define \(\mathrm {THH}\)  of a \(-enriched \)\(-category as follows. Let \)\( be the category with finite cyclically ordered sets \) 0<1<...<n<0\(, for \) n0\(, as objects and maps that respect the cyclic structure as morphisms. (For more details see \cite [Definition 3.3]{berman2019thh}.) Let \)\(({O}_{thh})_\bullet : \Lambda \rightarrow \mathcal {P}({B}\mathrm {ypass}_S)\)\( be the functor that takes a cyclic set \) 0<1<...<n<0\( to the coproduct of representables\)\(\coprod _{X_0,... ,X_n \in S}(X_0,X_1,...,X_n,X_0),\)\(and let\)\({O}_{thh}=\mathrm {colim}_{\Delta ^{op}} ({O}_{thh})_\bullet \in \mathcal {P}({B}\mathrm {ypass}_S).\)\(We suppress the choice of the set \) S\( of objects from the notation \) (Othh)\( and \) Othh\(.\) As Berman explains, any enriched \(\infty \) -category \( {B}\mathrm {ypass}_S \rightarrow  extends to a colimit-preserving functor\)\(*: \mathcal {P}({B}\mathrm {ypass}_S) \rightarrow \)\(which allows us to formulate the following generalization of \) THH\(, providing a way of associating an object in \)  to any \(-enriched \)\(-category.\)
Definition 2.10 ([8]}) 

Let \( be a \) -enriched \(\infty \) -category. The topological Hochschild homology of \( is\)\(\mathrm {THH}( = *({O}_{thh}) \in \mathrm {Obj} \)\(\)
Tracing through the definitions, since \(*\)  preserves colimits, we see that \(\mathrm {THH}(\)  is the colimit of a cyclic diagram that at level \(n\)  is equivalent to
\(\coprod _{X_0,...,X_n \in S} X_0,...,X_n,X_0) \simeq \coprod _{X_0,..., X_n} X_0,X_1) \otimes ... \otimes X_n,X_0)\) 
because \( is symmetric monoidal,\)
Example 2.11 If \( (\mathrm {Sp}, \wedge )\) , the symmetric monoidal \(\infty \) -category of spectra with the smash product, then a \(-enriched \)\(-category \)  is a spectral \(\infty \) -category, and \(\mathrm {THH}(\)  is the \(\infty \) -categorical analogue of Def:thh of spectral cat.

Example 2.12 If \(({S},\times )\) , the symmetric monoidal \(\infty \) -category of spaces, then \(-enriched \)\(-categories are preciselynon-enriched \)\(-categories. More concretely, we get complete Segal spaces \cite {rezk2001css}, as follows from \cite [Theorem 4.4.7, Remark 5.3.10]{gepnerhaugseng2015enriched}\)
Example 2.13 
If \( ({C}\mathrm {at}_\infty , \times )\) , then we call a \(-enriched \)\(-category \)  an \((\infty ,2)\) -category. Note that if \( is an \) (,2)\(-category, then \) THH(\( is an \)\(-category.\)
Remark 2.14 There are many models of \((\infty ,2)\) -categories in the literature, such as 2-complicial sets [13]}, 2-fold complete Segal spaces [14]}, and \(\Theta _2\) -spaces [15]}.
The model of \((\infty ,2)\) -categories constructed in ex:infty two cats corresponds to 2-fold complete Segal spaces, as follows from the proof in [6]}, with the \(\infty \) -category of complete Segal spaces \({C}{S}{S}\)  replacing the \(\infty \) -category \({S}\) .

Let \( be a \) -enriched \(\infty \) -category with set of objects \(S\) , and let \(\tau : \mathrm {THH}(\rightarrow V\)  be a morphism in \(. From \)\( we can derive a family of morphisms in \)
\(\lbrace \tau _{X}: X, X) \rightarrow V\mid X\in S\rbrace .\) 
The cyclic structure of \(({O}_{thh})_\bullet \)  implies that
\({X_0,X_1)\otimes X_1, X_0)\simeq X_0,X_1,X_0)[r] @<65pt> [dd]& X_0,X_0)[d]^{\tau _{X_0}}\\&V\\X_1,X_0)\otimes X_0, X_1)\simeq X_1,X_0,X_1)[r] &X_1,X_1)[u]_{\tau _{X_1}}}\) 
commutes for all \(X_0,X_1\in S\) , where the horizontal “composition” maps arise from bypasses, and the lefthand vertical arrow is given by the cyclic structure. In other words, the value of \(\tau \)  on a composite of two 1-cells that can be composed in either order is independent of the order of composition, just as the trace of a product of two matrices that can be multiplied in either order is independent of the order of multiplication. This observation justifies the terminology in the next definition.
Definition 2.15 
Let \( be a \) -enriched \(\infty \) -category, and let \(V\)  be an object in \(. A {\it trace} out of \)  into \(V\)  is a morphism \(\mathrm {THH}( \rightarrow V\)  in \(.\)
Shadows vs. traces
In this section we compare the point-set and \(\infty \) -categorical approaches to THH and deduce an equivalence between shadows on a bicategory \({B}\)  with values in a category \( and traces out of \) B\( into \) . To make sense of this comparison, we need in particular to explain how to define a category that can reasonably be called the topological Hochschild homology of \({B}\) .
Remark 3.1 Motivated by the analysis in subsec:thh of enriched categories and in order to simplify notation, for a given \(-enriched \)\(-category \) , we abuse notation and write
\(X_0,X_1,...,X_n,X_0) = X_0,X_1) \otimes X_1,X_2) ... \otimes X_n,X_0).\) 


\(\mathrm {THH}\)  of bicategories
Since bicategories are \((2,2)\) -categories, i.e., categories enriched in categories [6]}, it follows from [8]} that a bicategory is a \({C}\mathrm {at}\) -enriched category in the sense of def:enriched infty cat, where \({C}\mathrm {at}\)  is equipped with its Cartesian symmetric monoidal structure. We can therefore apply def:thh of enriched cat to associate to any bicategory \({B}\)  a category \(\mathrm {THH}({B})\) , which is its topological Hochschild homology.
The following, more explicit description of \(\mathrm {THH}({B})\)  proves useful in the next subsection.
Proposition 3.2 
For every bicategory \({B}\) , the category \(\mathrm {THH}({B})\)  is equivalent to the colimit of
\( \begin{tikzcd}[column sep=0.3in]\displaystyle \coprod _{X_0} {B}(X_0,X_0)[r, shorten >=1ex,shorten <=1ex]&\displaystyle \coprod _{X_0,X_1} {B}(X_0,X_1,X_0)[l, shift left=1.2, "d_1"] [l, shift right=1.2, "d_0"^{\prime }][r, shift right, shorten >=1ex,shorten <=1ex ] [r, shift left, shorten >=1ex,shorten <=1ex]&\displaystyle \coprod _{X_0,X_1,X_2} {B}(X_0,X_1,X_2,X_0)[l] [l, shift left=2, "d_2"] [l, shift right=2, "d_0"^{\prime }]\end{tikzcd}\) 

in \({C}\mathrm {at}\) .

Since \({C}\mathrm {at}\)  is a \((2,1)\) -category, for any simplicial diagram \(F: \Delta ^{op} \rightarrow {C}\mathrm {at}\) , there is an equivalence
\(\mathrm {colim}( \Delta ^{op} \xrightarrow{} {C}\mathrm {at}) \simeq \mathrm {colim}( (\Delta _{\le 2})^{op} \xrightarrow{} \Delta ^{op} \xrightarrow{} {C}\mathrm {at})\) 
where \(i: \Delta _{\le 2} \rightarrow \Delta \)  is the natural inclusion map of the full subcategory with objects \([0],[1]\)  and \([2]\)  [19]}.
Remark 3.4 
While it is not yet proven in full generality that the construction of \(\mathrm {THH}\)  (def:thh of enriched cat) is functorial, the explicit formula in eq:thh bicat implies that when \( ({C}\mathrm {at},\times )\) , the construction of \(\mathrm {THH}\)  is indeed functorial.

Remark 3.5 In order to define \(\mathrm {THH}\)  of a bicategory, none of the definitions in subsec:thh of enriched categories is actually needed: we could simply have defined it to be the colimit of def:thh of enriched cat
However, in sec:traces of infty two categories, when we compare \(\mathrm {THH}\)  of bicategories and of \((\infty ,2)\) -categories, the more abstract description proves necessary.

Notation 3.6 
Let \({B}\)  be a bicategory. For objects \(B_1, B_2\)  chosen among
\(\coprod _{X_0}{B}(X_0,X_0),\coprod _{X_0,X_1}{B}(X_0,X_1,X_0),\coprod _{X_0,X_1,X_2}{B}(X_0,X_1,X_2,X_0)\) 
we use \(\mathrm {Fun}^\Delta (B_1,B_2)\)  to denote the subcategory of \(\mathrm {Fun}(B_1,B_2)\)  consisting of functors and natural isomorphisms generated by those in eq:thh bicat.

Part of the data of a bicategory are explicit composition maps, which one can view as giving rise to the face maps \(d_i\)  in def:thh of enriched cat. In particular, again abusing notation somewhat, \(d_0,d_1: {B}(X_0,X_1,X_0) \rightarrow {B}(X_i,X_i)\)  (where \(i=0,1\) ) are given by
\(d_0(F,G) = F G, \ d_1(F,G) = G F.\) 
Moreover, the face maps on \(X_0,X_1,X_2,X_0)\)  are given by
\(d_0(F,G,H)=(FG,H), \ d_1(F,G,H)=(F,GH), \ d_2(F,G,H)=(HF,G)\) 
This explicit structure plays an important role in later sections.
It is important to note that the colimit of eq:thh bicat is evaluated in the \((2,1)\) -category \({C}\mathrm {at}\) , which differs from the 1-categorical colimit in the category of categories, but rather corresponds to the pseudocolimit [20]}. Computing general pseudocolimits can be quite challenging, although there certain helpful methods via weighted colimits of enriched categories [21]} and homotopy colimits of model categories [22]}. The example below illustrates the challenges that arise even in simple examples.
Example 3.7 
Let \((\otimes ,I)\)  be a monoidal category, and let \(B be the bicategory with one object and morphism category given by \) , with composition given by the monoidal product. Interpreting eq:thh bicat in this particular case, we see that \(\mathrm {THH}(B\)  is the colimit of the diagram
[column sep=0.3in]
[r, shorten >=1ex,shorten <=1ex]

[l, shift left=1.2, "d1"] [l, shift right=1.2, "d0"']
[r, shift right, shorten >=1ex,shorten <=1ex ] [r, shift left, shorten >=1ex,shorten <=1ex]

[l] [l, shift left=2, "d2"] [l, shift right=2, "d0"']
.
Although this diagram is easy to describe explicitly, there is no easy way to compute its colimit. In general the only description available of the category \(\mathrm {THH}(B\)  is formulated in terms of generators and relations [23]}, which is generally not useful for computations.

One case in which we can more easily compute \(\mathrm {THH}\)  of a bicategory, using homotopy theory, arises when the bicategory is actually a \((2,0)\) -category, i.e., all 1-cells and 2-cells are equivalences. We call such a bicategory a bigroupoid and denote the (3,1)-category of bigroupoids by \({G}\mathrm {rpd}_2\) . To compute \(\mathrm {THH}\)  of a bigroupoid, we need to understand the relationship between bigroupoids and spaces.
Remark 3.8 Let \({S}\)  be the \(\infty \) -category of spaces. By [6]}, there is an adjunction of \(\infty \) -categories
\( \begin{tikzcd}[row sep=0.5in, column sep=0.5in]{S}[r, shift left=1.8, "\Pi _{\le 1}", "\bot "^{\prime }] & {G}\mathrm {rpd}, [l, shift left=1.8, "N"]\end{tikzcd}\) 
where \({G}\mathrm {rpd}\)  denotes the \((2,1)\) -category of groupoids,
of which the right adjoint is fully faithful, with essential image given by 1-truncated spaces (spaces with trivial homotopy groups above degree 1). The right adjoint takes a groupoid \({G}\)  to a Kan complex of which the 0-cells are the objects of \({G}\) , while for any two objects \(X,Y\) , there is an isomorphism of sets \(\mathrm {Map}_{N{G}}(X,Y) \cong {G}(X,Y)\) , where \(\mathrm {Map}_{N{G}}\)  denotes the path space.
The same corollary ([6]}) implies that there is an adjunction
\( \begin{tikzcd}[row sep=0.5in, column sep=0.5in]{S}[r, shift left=1.8, "\Pi _{\le 2}", "\bot "^{\prime }] & {G}\mathrm {rpd}_2 [l, shift left=1.8, "N_2"]\end{tikzcd}\) 
of which the right adjoint is fully faithful, with essential image given by 2-truncated spaces. By [6]} the functor \(N_2\)  takes a bigroupoid \({G}\)  to a Kan complex \(N_2{G}\) , of which the 0-cells are the objects of \({G}\) , while for any two objects \(X,Y\) , there is an equivalence of groupoids
\(\Pi _1\mathrm {Map}_{N_2{G}}(X,Y) \simeq {G}(X,Y).\) 


We can use the adjunction above to compute \(\mathrm {THH}\)  of a bigroupoid.
Example 3.11 
For any bigroupoid \({G}\) , there is an equivalence
\(\mathrm {THH}({G}) = |\coprod _{X_0,...,X_n} {G}(X_0,X_1,...,X_0)| \simeq |\coprod _{X_0,...,X_n} \Pi _1\mathrm {Map}_{N_2{G}}(X_0,X_1,...,X_0)|\) 

\( \simeq \Pi _1 |\coprod _{X_0,...,X_n} \mathrm {Map}_{N_2{G}}(X_0,X_1,...,X_0)|\) 
where \(\mathrm {Map}_{N_2{G}}(X_0,X_1,...,X_0)\)  denotes the space of loops through the vertices \(X_0,X_1,...,X_n\)  and the equivalence follows from the previous remark and the fact that \(\Pi _1\)  commutes with colimits. The geometric realization of this diagram of spaces is known as unstable \(\mathrm {THH}\)  and has been computed to be the free loop space \((N_2{G})^{S^1}\)  [27]}. Since \(N_2\)  is fully faithful and \(N_2 B\mathbb {Z} = S^1\) , it follows that
\(\mathrm {THH}({G}) \simeq (N_2{G})^{S^1} = (N_2{G})^{N_2B\mathbb {Z}} \simeq \mathrm {Fun}({B}\mathbb {Z},{G}),\) 
where the last equivalence follows from the fact that \(N_2\)  is fully faithful.

Returning to the general case, we can use the explicit colimit presentation of \(\mathrm {THH}({B})\)  (eq:thh bicat) to better describe the category \(\mathrm {Fun}\big (\mathrm {THH}({B}),)\)  of traces out of a bicategory \({B}\)  into a category \(. Let \) ((2)op)\( be the join of \) (2)op\( with the terminal category. Intuitively \) ((2)op)\( is the category \) (2)op\( equipped with a new terminal object. Note there are evident inclusion functors \) (2)op ((2)op)\( and \) [0] ((2)op)\(, sending the unique object \) 0\( to the terminal object we denote by \) f\(.\)
Definition 3.12 
Let \({B}\)  be a bicategory and \( a category. Let \) B*: P(BypassB0)Cat\( be the colimit-preserving functor extending the functor \) BypassB0Cat\( encoding the bicategory structure of \) B\(.\) The category \({C}\mathrm {ocone}({B},\)  of \({B}\) -cocones taking values in \(\)  has as objects the pseudofunctors \(((\Delta _{\le 2})^{op})^{\rhd } \rightarrow {C}\mathrm {at}\)  that fit into the diagram
(2)op [d, hookrightarrow] [dr, "B*Othh"]
((2)op) [r, dashed]  Cat
{f} [ur, "'] [u, hookrightarrow]
and pseudonatural transformations as morphisms.

Concretely we can depict an object in \({C}\mathrm {ocone}({B},\)  as a diagram in \({C}\mathrm {at}\)  of the form
\( \begin{tikzcd}[row sep=0.3in, column sep=0.3in]\displaystyle \coprod _{X_0} {B}(X_0,X_0)[r, shorten >=1ex,shorten <=1ex] [dr, "C_0"^{\prime }]&\displaystyle \coprod _{X_0,X_1} {B}(X_0,X_1,X_0)[l, shift left=1.2, "d_1"] [l, shift right=1.2, "d_0"^{\prime }][r, shift right, shorten >=1ex,shorten <=1ex ] [r, shift left, shorten >=1ex,shorten <=1ex] [d, "C_1"]&\displaystyle \coprod _{X_0,X_1,X_2} {B}(X_0,X_1,X_2,X_0)[l] [l, shift left=2, "d_2"] [l, shift right=2, "d_0"^{\prime }] [dl, "C_2"]\\& \end{tikzcd}\) 
and a morphism as a commutative diagram of natural transformations of the form
\( \begin{tikzcd}[row sep=0.8in, column sep=0.3in]\displaystyle \coprod _{X_0} {B}(X_0,X_0)[r, shorten >=1ex,shorten <=1ex] [dr, "C_0^{\prime }"^{\prime }, bend right=20, ""{name=A}] [dr, "C_0" ""{name=U, below}, bend left=20]&\displaystyle \coprod _{X_0,X_1} {B}(X_0,X_1,X_0)[l, shift left=1.2, "d_1"] [l, shift right=1.2, "d_0"^{\prime }][r, shift right, shorten >=1ex,shorten <=1ex ] [r, shift left, shorten >=1ex,shorten <=1ex] [d, "C_1^{\prime }"^{\prime } very near start, bend right=20, ""{name=B}] [d, "C_1" very near start, bend left=20, ""{name=V, below}]&\displaystyle \coprod _{X_0,X_1,X_2} {B}(X_0,X_1,X_2,X_0).[l] [l, shift left=2, "d_2"] [l, shift right=2, "d_0"^{\prime }] [dl, "C_2^{\prime }"^{\prime } near start, bend right=20, ""{name=C}] [dl, "C_2", bend left=20, ""{name=W, below}]\\& [from=U, to=A, Rightarrow, "\alpha _0"^{\prime }, shorten <= 0.1in, shorten >= 0in][from=V, to=B, Rightarrow, "\alpha _1"^{\prime } near end, shorten <= 0.05in, shorten >= 0.0in][from=W, to=C, Rightarrow, "\alpha _2"^{\prime }, shorten <= 0.1in, shorten >= 0.1in]\end{tikzcd}.\) 
Since the objects in \({C}\mathrm {ocone}({B},\)  are pseudofunctors, the commutativity in eq:cocone and eq:cocone morphism holds up to appropriate choices of natural isomorphisms. Understanding the precise data of objects in \({C}\mathrm {ocone}({B},\)  is an important aspect of the proof of thm:main theorem. As this description is quite technical, it has been relegated to sec:pseudo diag.
Remark 3.15 For every bicategory \({B}\) , there is a \({B}\) -cocone taking values in \(\mathrm {THH}({B})\) , given by the maps into the colimit. As we observe below, this \({B}\) -cocone is universal, in the sense that every other \({B}\) -cocone factors through it.

The identification below follows immediately from the colimit description of \(\mathrm {THH}({B})\)  (eq:thh bicat) and the universal property of colimits.
Proposition 3.16 
The functor
\(\mathrm {Comp}:\mathrm {Fun}\big (\mathrm {THH}({B}),) \xrightarrow{}{C}\mathrm {ocone}({B}, ,\) 
which composes a functor \(\mathrm {THH}({B}) \rightarrow  with the universal \) B\(-cocone taking values in \) THH(B)\(, is an equivalence of categories, which is natural in \) B\( and \) .

In other words, traces out of \({B}\)  into \( ({defn:trace}) are equivalent to \) B\(-cocones taking values in \) .

The equivalence between shadows and traces
We are finally ready to establish the main result of this article, that shadows (subsec:shadows) are traces of bicategories (subsec:thh of enriched categories). To state the result precisely, we first introduce a notion of morphisms between shadows.
Definition 3.17 
Let \({B}\)  be a bicategory and \( a category. The {\it category of shadow functors from {B} to }, denoted\) Sha(B,\(, is specified as follows.\begin{enumerate}\item The objects are shadows, i.e., pairs \big (\langle \langle - \rangle \rangle ,\theta \big ) satisfying the conditions of {Def:shadow}.\item Given two shadows \big (\langle \langle - \rangle \rangle _1, \theta _1\big ), \big (\langle \langle - \rangle \rangle _2, \theta _2\big ), a morphism between them consists of a natural transformation\alpha _{-}: \langle \langle - \rangle \rangle _1 \rightarrow \langle \langle - \rangle \rangle _2such that for any pair of 1-morphisms F: X_0 \rightarrow X_1, G: X_1 \rightarrow X_0 in {B}, the diagram\begin{equation} \begin{tikzcd}[row sep=0.5in, column sep=0.5in]\langle \langle FG \rangle \rangle _1 [r, "\theta _1"] [d, "\alpha _{FG}"] & \langle \langle GF \rangle \rangle _1 [d, "\alpha _{GF}"] \\\langle \langle FG \rangle \rangle _2 [r, "\theta _2"] & \langle \langle GF \rangle \rangle _2,\end{tikzcd}\end{equation}in  commutes, and \alpha  commutes with associators and unitors as well.\end{enumerate}\)
Theorem 3.18 
For any bicategory \({B}\)  and category \(, there is an equivalence of categories\)\(\mathrm {Fun}\big (\mathrm {THH}({B}),) \xrightarrow{} {S}\mathrm {ha}({B},\)\(that factors via the functor \) Comp\( ({lemma:cocones are traces}) through the category \) Cocone(B,\( ({def:cocone}) and that is natural in \) B\( and \) . Here \({S}\mathrm {ha}({B},\)  denotes the category of shadow functors from \({B}\)  to \( ({def:category of shadows}).\)
The proof is quite long, so we break it down into steps.
Step (I) Reduction to \({S}\mathrm {t}\)  and \({U}\mathrm {n}\) :
By lemma:cocones are traces the functor
\(\mathrm {Comp}: \mathrm {Fun}\big (\mathrm {THH}({B}),) \rightarrow {C}\mathrm {ocone}({B},\) 
is a natural equivalence of categories. In order to prove the theorem, we define two functors natural in \({B}\)  and \( that we think of as performing ``strictification" and ``unstrictification" processes ({rmk:strictification}),\)\({S}\mathrm {t}: {C}\mathrm {ocone}({B}, \rightarrow {S}\mathrm {ha}({B},\)\(and\)\({U}\mathrm {n}: {S}\mathrm {ha}({B}, \rightarrow {C}\mathrm {ocone}({B},,\)\(and then prove that they are mutually inverse.\medskip \)
Step (II) The functor \({S}\mathrm {t}\)  on objects:
For any cocone \(C = (C_0,C_1,C_2)\)  in \({C}\mathrm {ocone}({B},\) , the functor underlying \({S}\mathrm {t}(C)\)  is
\(C_0: \coprod _{X_0} {B}(X_0,X_0) \rightarrow \) 
The natural isomorphism witnessing the trace-like property of \(C_0\)  is the composite of the natural isomorphisms witnessing the commutativity of the lefthand triangles in eq:cocone,
\(\theta _C: C_0d_0 \xrightarrow{} C_1 \xrightarrow{} C_0d_1.\) 
Since this assignment is clearly natural in \({B}\)  and \(, what remains is to prove that \) (C0,C)\( satisfies the axioms of a shadow, i.e., that the two diagrams of {Def:shadow} commute. First we show that {eq:shadow cond one} commutes.\) The fact that eq:cocone is a cocone means that the image of the map

\(\displaystyle C_0: \mathrm {Fun}^\Delta \big (\coprod _{X_0,X_1,X_2} {B}(X_0,X_1,X_2,X_0), \coprod _{X_0} {B}(X_0,X_0)\big ) \rightarrow \mathrm {Fun}\big (\coprod _{X_0,X_1,X_2} {B}(X_0,X_1,X_2,X_0),)\) 

must be contractible (here we are using not:simplicial maps). Concretely, this means the image under postcomposition with \(C_0\)  of any two functors \(\coprod _{X_0,X_1,X_2} {B}(X_0,X_1,X_2,X_0) \rightarrow \coprod _{X_0} {B}(X_0,X_0)\)  induced by the simplicial operators must be naturally isomorphic in a unique way (as a non-unique natural isomorphism would give us a non-trivial loop).
eq:cocone gives rise to the cube below, in which each face commutes up to natural isomorphism.
 
[row sep=0.3in, column sep=0.3in]
[-0.3in]  X0,X1,X2 B(X0,X1,X2,X0)   [0.3in]  X0,X1 B(X0,X1,X0)

X0,X1 B(X0,X1,X0)     X0B(X0,X0)

  X0,X1B(X0,X1,X0)     X0B(X0,X0)

X0B(X0,X0)      
[from=3-1, to=5-3, Rightarrow,"a" description, dashed]
[from=5-3, to=1-7, Rightarrow,"a" description, dashed, bend left=10]
[from=7-1, to=5-7, Rightarrow, "" description, dashed]
[from=3-5, to=5-7, Rightarrow, "" description, dashed]
[from=1-3, to=1-7, "d2" description]
[from=1-3, to=5-3, "d1" description, crossing over]
[from=1-3, to=3-1, "d0"' description]
[from=1-7, to=3-1, Rightarrow,"a"' description, dashed, crossing over]
[from=1-7, to=5-7, "d1" description]
[from=1-7, to=3-5, "d0" description]
[from=3-1, to=3-5, "d1" description, crossing over]
[from=3-1, to=7-1, "d0" description]
[from=5-3, to=5-7, "d1" description]
[from=5-3, to=7-1, "d0" description]
[from=7-1, to=3-5, Rightarrow, "" description, dashed, bend right=8, crossing over]
[from=3-5, to=7-5, "C0" description, crossing over]
[from=5-7, to=7-5, "C0" description]
[from=7-1, to=7-5, "C0" description]

There are exactly six paths in this cube from the top left corner to the bottom right corner,
\(\displaystyle \coprod _{X_0,X_1,X_2} {B}(X_0,X_1,X_2,X_0) \rightarrow \) 
corresponding to the six objects in eq:shadow cond one. The natural isomorphisms on the faces of the cube correspond to the morphisms between the objects in eq:shadow cond one. As explained above, the diagram of natural isomorphisms must commute since eq:cocone is a cocone.
It remains to prove that eq:shadow cond two commutes. By symmetry it suffices to verify commutativity of the left-hand triangle, which we do by an argument similar to that above. Since eq:cocone is a cocone, the image of
\(C_0:\mathrm {Fun}^\Delta \big (\coprod _{X_0}{B}(X_0,X_0),\coprod _{X_0}{B}(X_0,X_0)\big ) \rightarrow \mathrm {Fun}\big (\coprod _{X_0}{B}(X_0,X_0) ,)\) 
must be contractible, which means that any two functors from \({B}(X_0,X_0)\)  to \( induced by the simplicial diagram in the domain functor category must be natural isomorphic in a unique manner.\) Since the morphisms in the left-hand triangle in eq:shadow cond two are exactly the natural isomorphisms in the diagram below, we can conclude by the remark above that the triangle commutes as desired.
[row sep=0.5in, column sep=0.5in]
X0 B(X0,X0) [dr, "s0"] [ddr, "id"', bend right=30, ""name=U, below] [drr, "id", bend left=20, ""name=A]
 X0,X1 B(X0,X1,X0) [r, "d1"] [d, "d0"]
 X0 B(X0,X0) [d, "C0"]
 X0 B(X0,X0) [r, "C0"]  [from=2-2, to=U, Rightarrow, "r"', shorten=0.1in]
[from=2-2, to=A, Rightarrow, "l", shorten=0.2in]
[from=3-2, to=2-3, Rightarrow, "", shorten=0.2in]
Step (III) The functor \({S}\mathrm {t}\)  on morphisms:
We now explain how to define \({S}\mathrm {t}\)  on morphisms and show that it is indeed a functor.
Let
\((\alpha _0,\alpha _1,\alpha _2): (C_0,C_1,C_2) \Rightarrow (C_0^{\prime },C_1^{\prime },C_2^{\prime })\) 
be a morphism of cocones. We just proved that the functors \(C_0\)  and \( C_0^{\prime }\)  can be equipped with natural transformations \(\theta \)  and \(\theta ^{\prime }\)  with respect to which they are shadows. By definition
\(\alpha _0: C_0 \rightarrow C_0^{\prime }\) 
is a natural transformation. We need to check that \(\alpha _0\)  satisfies the conditions formulated in def:category of shadows with respect to \(\theta \)  and \(\theta ^{\prime }\) , allowing us to set \({S}\mathrm {t}(\alpha _0,\alpha _1,\alpha _2)=\alpha _0\) . Preservation of composition and of identities by
\({S}\mathrm {t}\)  is then obvious.
The compatibility of \(\alpha _0\)  and \(\alpha _1\)  with the natural isomorphisms in the cocones \(C\)  and \(C^{\prime }\)  implies that the diagram
[row sep=0.5in, column sep=0.5in]
C0d0 [rr, "", bend left = 30] [d, "0"]  C1 [d, "1"] [r, ""] [l, ""']  C0d1 [d, "0"]
C0'd0 [rr, "'"', bend right = 30]  C1' [r, ""'] [l, ""]  C0'd1
commutes and hence that eq:shadow cat cond one commutes. It remains to show that \(\alpha _0\)  commutes with associators and unitors, which follows from the diagrams below, by standard whiskering arguments in bicategories, again using, as in the previous step, that \((C_0,C_1, C_2)\)  and \((C^{\prime }_0,C^{\prime }_1, C^{\prime }_2)\)  are cocones.

[row sep=0.3in, column sep=0.3in]
 X0,X1,X0 B(X0,X1,X0) [dr, "d0", bend left=10] [dd, Rightarrow, "a" description, shorten =0.2in]  [-0.45in]
X0,X1,X2 B(X0,X1,X2,X0) [ur, "d0", bend left=10] [dr, "d1", bend right=10]  
X0 B(X0,X0)  [rrr, bend left = 40, ""name=B, below, "C0"] [rrr, bend right = 40, ""name=U, "C0'"'] [0.4in]   
 X0,X1,X0 B(X0,X1,X0) [ur, "d0", bend right=10] 
[from=B, to=U, Rightarrow, "0" description]
[row sep=0.3in, column sep=0.3in]
 X0,X1 B(X0,X1,X0) [dr, "d0"]  [-0.45in]
X0 B(X0,X0) [rr, "Id" near start, ""name=C, ""name=D, below] [ur, "s0"] [dr, "s0"']   X0 B(X0,X0)  [rrr, bend left = 40, ""name=B, below, "C0"] [rrr, bend right = 40, ""name=U, "C0'"'] [0.4in]   
 X0,X1 B(X0,X1,X0) [ur, "d1"'] 
[from=1-2, to=C, Rightarrow, "r" description]
[from=3-2, to=D, Rightarrow, "l" description]
[from=B, to=U, Rightarrow, "0" description]
Thus \(\alpha _0\)  is indeed a morphism of shadows.
Step (IV) The functor \({U}\mathrm {n}\)  on objects:
Given any shadow functor \(\big (\langle \langle - \rangle \rangle , \theta \big )\)  from \({B}\)  to \(, we construct a \) B\(-cocone taking values in \) , using lemma:trunc simpl lift in the case where the target 2-category is \({C}\mathrm {at}\) , and the functor \(F\)  is \({O}_{thh}\) . We need thus to choose an appropriate category, functor, and natural isomorphism that satisfy the conditions stated in lemma:trunc simpl lift.
The extra condition on the extension in the definition of \({B}\) -cocones (def:cocone) implies that the category chosen as the image of the terminal object must be \(. The functor we choose as the image of \) t0:0f\( is \) - : X0 B(X0,X0) , while the natural isomorphism chosen should clearly be that given as part of the shadow structure, \(\theta : \langle \langle FG \rangle \rangle  \cong \langle \langle GF \rangle \rangle \) .
We need to prove that the two equalities of 2-cells stated in lemma:trunc simpl lift hold for the choices that we have made.
The first condition in lemma:trunc simpl lift translates to the following diagram
\( \begin{tikzcd}[row sep=0.5in, column sep=0.5in]\langle \langle d_0s_0(-) \rangle \rangle  [r, "\theta "] [dr, "\langle \langle r \rangle \rangle "^{\prime }] & \langle \langle d_1s_0(-) \rangle \rangle  [d, "\langle \langle l \rangle \rangle "] \\& \langle \langle - \rangle \rangle \end{tikzcd},\) 
which commutes by eq:shadow cond two.
The second condition that the natural isomorphisms in lemma:trunc simpl lift must satisfy translates to the commutativity of the following diagram.
\( \begin{tikzcd}[row sep=0.5in, column sep=0.5in]\langle \langle d_0d_0(-) \rangle \rangle  [r, "\theta "] [d, "\langle \langle a \rangle \rangle "^{\prime }] & \langle \langle d_1d_0(-) \rangle \rangle  [r, "\langle \langle a \rangle \rangle "] & \langle \langle d_0d_2(-) \rangle \rangle  \\\langle \langle d_0d_1(-) \rangle \rangle  [r, "\theta "] & \langle \langle d_1d_1(-) \rangle \rangle  [r, "\langle \langle a \rangle \rangle "] & \langle \langle d_1d_2(-) \rangle \rangle  [u, "\theta "^{\prime }]\end{tikzcd}\) 
Since the commutativity of this diagram is exactly eq:shadow cond one in the definition of a shadow, we can conclude.
Step (V) The functor \({U}\mathrm {n}\)  on morphisms:
We now extend \({U}\mathrm {n}\)  to a functor. Let
\(\alpha : \langle \langle - \rangle \rangle _1 \rightarrow \langle \langle - \rangle \rangle _2\) 
be a morphism in \({S}\mathrm {ha}({B},\) , i.e., a natural transformation such that the diagrams in def:category of shadows commute. Our goal is it to construct a natural transformation from \({U}\mathrm {n}(\langle \langle - \rangle \rangle _1) \rightarrow {U}\mathrm {n}(\langle \langle - \rangle \rangle _2)\) , which informally should have following shape.
[row sep=1.5in, column sep=1in]
X0 B(X0,X0)
[r, shorten >=1ex,shorten <=1ex] [dr, "- 2"' very near start, bend right=15, ""name=A] [dr, "- 1" very near start, ""name=U, below, bend left=15]

X0,X1 B(X0,X1,X0)
[l, shift left=1.2, "d1"] [l, shift right=1.2, "d0"']
[r, shift right, shorten >=1ex,shorten <=1ex ] [r, shift left, shorten >=1ex,shorten <=1ex] [d, "d0(-) 2"' very near start, bend right=17, ""name=B] [d, "d0(-) 1" very near start, bend left=17, ""name=V, below]

X0,X1,X2 B(X0,X1,X2,X0).
[l] [l, shift left=2, "d2"] [l, shift right=2, "d0"'] [dl, "d0d0(-) 2"' very near start, bend right=10, ""name=C] [dl, "d0d0(-) 1" very near start, bend left=10, ""name=W, below]
 
[from=U, to=A, Rightarrow, ""', shorten <= 0.1in, shorten >= 0in]
[from=V, to=B, Rightarrow, "d0"' near end, shorten <= 0.05in, shorten >= 0.0in]
[from=W, to=C, Rightarrow, "d0d0"', shorten <= 0.1in, shorten >= 0.1in]
More precisely, a natural transformation \({U}\mathrm {n}(\langle \langle - \rangle \rangle _1) \rightarrow {U}\mathrm {n}(\langle \langle - \rangle \rangle _2)\)  is a pseudofunctor
\({U}\mathrm {n}\alpha : ((\Delta _{\le 2})^{op})^{\rhd } \rightarrow \mathrm {Fun}([1],{C}\mathrm {at})\) 
that fits into the following diagram.
\(\begin{tikzcd}[column sep=0.6in]& & {C}\mathrm {at}\\(\Delta _{\le 2})^{op} [r] [rr, bend right = 20, "\mathrm {id}"^{\prime } near start] &((\Delta _{\le 2})^{op})^{\rhd } [r, dashed, "{U}\mathrm {n}\alpha " description] [ur, "{U}\mathrm {n}\langle \langle - \rangle \rangle _1"near end] [dr, "{U}\mathrm {n}\langle \langle - \rangle \rangle _2"^{\prime } near end]& \mathrm {Fun}([ 1 ] ,{C}\mathrm {at}) [u, "\pi _1"^{\prime }] [d, "\pi _2"] \\& & {C}\mathrm {at}\end{tikzcd}\) 
Here, the arrow labeled \(\mathrm {id}\)  corresponds the identity natural transformation on the functor \({B}_*\circ {O}_{thh}\)  (see def:cocone), which is the common value of the restrictions of \({U}\mathrm {n}\langle \langle - \rangle \rangle _1\)  and \({U}\mathrm {n}\langle \langle - \rangle \rangle _2\)  to \((\Delta _{\le 2})^{op}\) .
Our goal is thus to construct such a functor \({U}\mathrm {n}\alpha \) .
To lift the functor \(\mathrm {id}:(\Delta _{\le 2})^{op} \rightarrow \mathrm {Fun}([1],{C}\mathrm {at})\) , we again use lemma:trunc simpl lift. We proceed now to make the necessary choices and verify the necessary conditions. For the remainder of the proof, it is helpful to keep in mind that \(\mathrm {Fun}([1],{C}\mathrm {at})\)  is a bicategory with functors \([1] \rightarrow {C}\mathrm {at}\)  as objects and squares that commute up to natural isomorphisms as morphisms.
Considering the hypotheses of lemma:trunc simpl lift, this time with \(\mathrm {Fun}([ 1 ] ,{C}\mathrm {at})\)  as target bicategory, we choose the image of the final object to be the identity functor on \( and the image of the 1-cell \) t0:0f\( to be \)\( itself, seen as filling in a square where the composites along the sides are \) id- 1\( and \) - 2 idB(X0,X0)\(. Given these choices, we must now find an appropriate natural isomorphism between \) d0\( and \) d1\(, for which we select the diagram\)
\(\begin{tikzcd}\langle \langle d_0(-) \rangle \rangle _1 [r, "\theta _1"] [d, "\alpha d_0"] & \langle \langle d_1(-) \rangle \rangle _1 [d, "\alpha d_1"] \\\langle \langle d_0(-) \rangle \rangle _2 [r, "\theta _2"] & \langle \langle d_1(-) \rangle \rangle _2\end{tikzcd}\) 
which commutes by the conditions of a morphism of shadows (def:category of shadows).
We now need to verify the two conditions imposed on the natural isomorphisms from lemma:trunc simpl lift. The first one requires that the following diagram commute
\( \begin{tikzcd}[row sep=0.4in, column sep=0.6in]& \langle \langle d_0s_0 \rangle \rangle _1 [rr, "\theta _1"] [dl, "\alpha d_0s_0" description] [ddr, "\langle \langle r \rangle \rangle _1" near start] & & \langle \langle d_1s_0 \rangle \rangle _1 [dl, "\alpha d_1s_0" description] [ddl, "\langle \langle l \rangle \rangle _1"] \\\langle \langle d_0s_0 \rangle \rangle _2 [rr, "\theta _2", crossing over] [ddr, "\langle \langle r \rangle \rangle _2"^{\prime }] & & \langle \langle d_1s_0 \rangle \rangle _2 [ddl, "\langle \langle l \rangle \rangle _2" description, crossing over] & \\& & \langle \langle - \rangle \rangle _1 [dl, "\alpha " description] & \\& \langle \langle - \rangle \rangle _2 & &\end{tikzcd},\) 
whereas the second condition requires the diagram below to commute.
\( \begin{tikzcd}[row sep=0.3in, column sep=0.1in]\langle \langle d_0d_0(-) \rangle \rangle _1 [dr, "\alpha d_0d_0" description] [rr, "\theta _1"] [dd, "\langle \langle a \rangle \rangle _1"^{\prime }] & & \langle \langle d_1d_0(-) \rangle \rangle _1 [dr, "\alpha d_1d_0" description] [rr, "\langle \langle a \rangle \rangle _1"] & & \langle \langle d_0d_2(-) \rangle \rangle _1 [dr, "\alpha d_0d_2" description] \\& \langle \langle d_0d_0(-) \rangle \rangle _2 [rr, "\theta _2"] [dd, "\langle \langle a \rangle \rangle _2"^{\prime } near start] & & \langle \langle d_1d_0(-) \rangle \rangle _2 & & \langle \langle d_0d_2(-) \rangle \rangle _2\\\langle \langle d_0d_1(-) \rangle \rangle _1 [rr, "\theta _1" near end] [dr, "\alpha d_0d_1" description] & & \langle \langle d_1d_1(-) \rangle \rangle _1 [rr, "\langle \langle a \rangle \rangle _1"] [dr, "\alpha d_1d_1" description] & & \langle \langle d_1d_2(-) \rangle \rangle _1 [uu, "\theta _1"^{\prime } near start] [dr, "\alpha d_1d_2" description]\\& \langle \langle d_0d_1(-) \rangle \rangle _2 [rr, "\theta _2"] & & \langle \langle d_1d_1(-) \rangle \rangle _2 [rr, "\langle \langle a \rangle \rangle _2"] & & \langle \langle d_1d_2(-) \rangle \rangle _2 [uu, "\theta _2"^{\prime }][from=2-4, to=2-6, "\langle \langle a \rangle \rangle _2" near start, crossing over]\end{tikzcd}\) 
To establish the commutativity of these two diagrams, we use that the outer squares and triangles in the diagram commute (by def:category of shadows), whence the inside diagram commutes as well, since categories are 2-coskeletal.
Applying lemma:trunc simpl lift, we to deduce that we have indeed defined a functor \({U}\mathrm {n}\alpha : ((\Delta _{\le 2})^{op})^\rhd \rightarrow \mathrm {Fun}([1],{C}\mathrm {at})\) , which fits by construction into eq:lift nat, since \(\pi _1{U}\mathrm {n}(\alpha )= {U}\mathrm {n}(\langle \langle - \rangle \rangle _1)\)  and \(\pi _2{U}\mathrm {n}(\alpha )= {U}\mathrm {n}(\langle \langle - \rangle \rangle _2)\) .
Step (VI): \({S}\mathrm {t}\)  and \({U}\mathrm {n}\)  are mutually inverse:
We now prove that \({S}\mathrm {t}\)  and \({U}\mathrm {n}\)  are inverse to each other.
If \(\big (\langle \langle - \rangle \rangle , \theta \big )\)  is a shadow, then
\({U}\mathrm {n}\Big (\langle \langle - \rangle \rangle ,\theta \Big ) = \Big (\langle \langle - \rangle \rangle ,\langle \langle d_0(-) \rangle \rangle , \langle \langle d_0d_0(-) \rangle \rangle \Big )\) 
and so
\({S}\mathrm {t}{U}\mathrm {n}\big (\langle \langle - \rangle \rangle ,\theta \big ) = \big (\langle \langle - \rangle \rangle ,\theta \big ),\) 
i.e., \({S}\mathrm {t}{U}\mathrm {n}\)  is the identity functor.
On the other hand, let \(C=(C_0,C_1,C_2)\)  be an arbitrary cocone. By cor:ps-2 this cocone (which is by definition a pseudofunctor) is naturally equivalent to a strict cocone, which, by lemma:trunc simpl lift is of the form \((C_0,C_0d_0,C_0d_0d_0)\) . By functoriality \({S}\mathrm {t}\)  and \({U}\mathrm {n}\)  preserve natural isomorphisms, so it suffices to prove that \({U}\mathrm {n}{S}\mathrm {t}\)  takes this particular cocone to itself.
This follows by direct computation. Indeed \({S}\mathrm {t}(C_0,C_0d_0,C_0d_0d_0)\)  is a shadow with shadow functor \(C_0\)  and \({U}\mathrm {n}\) , by its very definition, takes this shadow to the cocone \((C_0,C_0d_0,C_0d_0d_0)\) .
Remark 3.25 
The proof above can be thought of as a “strictification argument". We strictify a general diagram such that the total amount of data reduces to that of a shadow. The proof shows that no data is lost in this strictification process. A shadow can thus be characterized as the minimal amount of data required to describe a cocone.

Remark 3.26 
Applying thm:main theorem to the shadow on the bicategory \(\mathrm {Ho}({M}\mathrm {od})\)  of thm:cp gives rise to a trace (defn:trace) out of \(\mathrm {Ho}({M}\mathrm {od})\)  into \(\mathrm {Ho}(\mathrm {Sp})\) ,
\(\mathrm {THH}\big (\mathrm {Ho}({M}\mathrm {od})\big ) \rightarrow \mathrm {Ho}(\mathrm {Sp}).\) 


Morita Invariance of Shadows via \(\mathrm {THH}\)
An important property satisfied by \(\mathrm {THH}\)  of ring spectra is Morita invariance: if two ring spectra \(R\)  and \(R^{\prime }\)  are Morita equivalent (i.e., their model categories of modules are Quillen equivalent), then \(\mathrm {THH}(R) \simeq \mathrm {THH}(R^{\prime })\)  [28]}. In line with the idea that a shadow is an axiomatic, bicategorical generalization of \(\mathrm {THH}\) , Campbell and Ponto proved that shadows satisfied a natural generalization of Morita invariance [1]}. In this section we recover Morita invariance of shadows using the machinery we developed in the previous sections, which enables us to define a functorial lift of the useful Euler characteristic construction of Campbell and Ponto, whose work we begin by reviewing.
Let \({B}\)  be a bicategory equipped with a shadow functor \(\langle \langle - \rangle \rangle \)  taking values in a category \(. Given an adjunction in \) B\(, i.e., a diagram\begin{equation} \begin{tikzcd}[row sep=0.5in, column sep=0.5in]C [r, bend left = 40, "F"] [loop, in=130, out=225, looseness=10, ""{name=D, below}, "GF"] &D [l, bend left =40, "G"] [loop, in=45, out=315, looseness=10, ""{name=U}, "FG"^{\prime }][from=1-1, to=D, Rightarrow, "u" description][from=U, to=1-2, Rightarrow, "c" description]\end{tikzcd}\end{equation}of 0-, 1-, and 2-cells in \) B\( satisfying the triangle identities, the {\it Euler characteristic} \) (F)\( of the \) 1\(-cell \) F\( is defined to be the composite morphism\)\(\langle \langle U_C \rangle \rangle  \xrightarrow{} \langle \langle GF \rangle \rangle  \cong \langle \langle FG \rangle \rangle  \xrightarrow{} \langle \langle U_D \rangle \rangle \)\(in \) .
Morita invariance of shadows is formulated as follows in terms of the Euler characteristic.
Proposition 4.1 (
[1]}) 
If the adjunction in eq:adj in C is actually an equivalence, i.e., if the 2-cells \(c\)  and \(u\)  are invertible, then \(\chi (F)\)  is an isomorphism in \( with inverse given by \) (G)\(.\)
We provide an alternative proof of this result below, exploiting the equivalence between shadows and functors out of \(\mathrm {THH}\) , which leads to a formal, functorial construction of \(\chi (F)\) . By thm:main theorem, a shadow out of \({B}\)  is equivalent to a functor out of the category \(\mathrm {THH}({B})\) . We can therefore focus the proof of Morita invariance on the representing object.
To formulate the main result of this section, we require the following definitions and notation. Let \(\Delta _+\)  be the category of finite ordinals and order-preserving morphisms, which can also be characterized as the category \(\Delta \)  together with one additional, initial object corresponding to the empty ordinal. With this notation at hand, the free adjunction bicategory \({A}\mathrm {dj}\)  can be described as follows. It has two objects \(0,1\) , and the category of morphisms from 0 to 0 is \(\Delta _+\) , where an ordinal of cardinality \(n\)  corresponds to the composite functor \((GF)^n\)  in eq:adj in C. In particular, the empty ordinal corresponds to the identity map. Similarly, the category of morphisms from 1 to 1 is \((\Delta _+)^{op}\) , and there are similar descriptions of the category of morphisms from 0 to 1 and 1 to 0. For more details, see the original description of the free adjunction in [31]}.
Our proof of Morita invariance is based on understanding \(\mathrm {THH}({A}\mathrm {dj})\) . As \({A}\mathrm {dj}\)  is quite complicated, we study instead an approximation to \({A}\mathrm {dj}\) .
Definition 4.2 
Let \({A}\mathrm {dj}_{\le 0}\)  be the strict bicategory with two objects \(0,1\)  and
\({A}\mathrm {dj}_{\le 0}(0,0) = \lbrace  \mathrm {U}_0 \xrightarrow{} g \circ f \rbrace , \) 

\({A}\mathrm {dj}_{\le 0}(1,1) = \lbrace  f \circ g \xrightarrow{} \mathrm {U}_1 \rbrace , \) 

\({A}\mathrm {dj}_{\le 0}(0,1) = \lbrace  f \rbrace , \) 

\({A}\mathrm {dj}_{\le 0}(1,0) = \lbrace  g \rbrace . \) 
where we have suppressed identity 2-cells from the description of the hom-categories, and composition is specified in the only way possible. In particular, \((f\circ g) \circ f = f= f\circ (g\circ f)\)  and \((g\circ f) \circ g = g = g\circ (f\circ g)\) .

Remark 4.3 
The strict bicategory \({A}\mathrm {dj}_{\le 0}\)  can be depicted by the diagram
[row sep=0.5in, column sep=0.5in]
0 [r, bend left = 40, "f"] [loop, in=130, out=225, looseness=10, ""name=D, below, "g f"] 
1 [l, bend left =40, "g"] [loop, in=45, out=315, looseness=10, ""name=U, "f g"']
[from=1-1, to=D, Rightarrow, "u" description]
[from=U, to=1-2, Rightarrow, "c" description]
.

Our description of \({A}\mathrm {dj}\)  implies there is an evident quotient functor
\(\pi _{\le 0}: {A}\mathrm {dj}\rightarrow {A}\mathrm {dj}_{\le 0},\) 
which restricts on morphisms categories to the quotient map
\({A}\mathrm {dj}(0,0) = \Delta _+ \rightarrow (\Delta _+)_{\le 0} = {A}\mathrm {dj}_{\le 0}(0,0)\) 
giving us a justification why we have chosen the notation \({A}\mathrm {dj}_{\le 0}\) .
Computing \(\mathrm {THH}({A}\mathrm {dj}_{\le 0})\)  turns out to tell us everything we need to know about Morita invariance of shadows. To formulate the result of this computation, we require one further definition.
Definition 4.5 
Let \([2]_{Iso}\)  be the category with four objects \(\lbrace 0,1,1^{\prime },2\rbrace \)  and non-identity morphisms depicted as follows
\(0 \rightarrow 1 \xrightarrow{} 1^{\prime } \rightarrow 2,\) 
meaning there are unique arrows \(0 \le 1 \le 2\)  and \(0 \le 1^{\prime } \le 2\)  and that 1 and \(1^{\prime }\)  are isomorphic via a unique isomorphism.

Note that \([2]_{Iso}\)  is equivalent to the poset \([2] = \lbrace  0 \le 1 \le 2 \rbrace \) , which justifies the naming convention. The following colimit description of \([2]_{Iso}\)  is easy to check.
Lemma 4.6 
There is a colimit diagram in \({C}\mathrm {at}\)
[column sep=0.3in]
[0 ][0 ][r] [r, shift left=2] [r, shift right=2]

[0 ][r, shift left=1.2, "1"] [r, shift right=1.2, "1'"']
[l, shift right, shorten >=1ex,shorten <=1ex ] [l, shift left, shorten >=1ex,shorten <=1ex]

{0 1 } {1' 2}
[l, shorten >=1ex,shorten <=1ex] [r]

[2 ]Iso.

We can now take the key step towards proving Morita invariance of shadows, by relating \(\mathrm {THH}({A}\mathrm {dj}_{\le 0})\)  to the colimit diagram given in lemma:two iso colim.
Theorem 4.7 
The colimit diagram below induces an isomorphism of colimits

[column sep=0.2in, row sep=0.5in]
[0 ][0 ][d, "(f,id,g) (f,g ,id)"]
[r] [r, shift left=2] [r, shift right=2]

[0 ][d, "(f ,g)"]
[r, shift left=1.2, "1"] [r, shift right=1.2, "1'"']
[l, shift right, shorten >=1ex,shorten <=1ex ] [l, shift left, shorten >=1ex,shorten <=1ex]

{0 1 } {1' 2} [d, "u c"]
[l, shorten >=1ex,shorten <=1ex] [r]

[2 ]Iso [d, ""]
X0,X1,X2 Adj0(X0,X1,X2,X0)
[r] [r, shift left=2, "d2"] [r, shift right=2, "d0"']

X0,X1 Adj0(X0,X1,X0)
[r, shift left=1.2, "d1"] [r, shift right=1.2, "d0"']
[l, shift right, shorten >=1ex,shorten <=1ex ] [l, shift left, shorten >=1ex,shorten <=1ex]

Adj0(0,0) Adj0(1,1)
[l, shorten >=1ex,shorten <=1ex] [r]

THH(Adj0)

We prove that \(\mathrm {THH}({A}\mathrm {dj}_{\le 0})\)  has the appropriate universal property, by computing compute \(\mathrm {Fun}(\mathrm {THH}({A}\mathrm {dj}_{\le 0}),\)  for an arbitrary category \(. By {thm:main theorem}, there is an equivalence of categories\)\(\mathrm {Fun}(\mathrm {THH}({A}\mathrm {dj}_{\le 0}), \xrightarrow[ \ \simeq \ ]{ \ \mathrm {Comp}\ } {C}\mathrm {ocone}({A}\mathrm {dj}_{\le 0}, \xrightarrow[ \ \simeq \ ]{ \ {S}\mathrm {t}\ } {S}\mathrm {ha}({A}\mathrm {dj}_{\le 0},,\)\( hence our first step is to characterize shadows on the bicategory \) Adj0\(.\) Recall that
\(\coprod _{X_0\in ({A}\mathrm {dj}_{\le 0})_0 }{A}\mathrm {dj}_{\le 0}(X_0,X_0) = \Big \lbrace \begin{tikzcd}[row sep=0.1in, column sep=0.1in]\overset{fg}{\bullet } [d, "c"^{\prime }] \\\underset{\mathrm {U}_1}{\bullet }\end{tikzcd}, \begin{tikzcd}[row sep=0.1in, column sep=0.1in]\overset{\mathrm {U}_0}{\bullet } [d, "u"] \\\underset{gf}{\bullet }\end{tikzcd}\Big \rbrace .\) 
The data of a shadow \(\langle \langle - \rangle \rangle \)  on \({A}\mathrm {dj}_{\le 0}\)  with values \( is thus determined by two functors\)\( \lbrace fg \xrightarrow{} \mathrm {U}_1\rbrace  \rightarrow \text{and}\quad \lbrace \mathrm {U}_0 \xrightarrow{} gf\rbrace  \rightarrow \)  and a natural isomorphism \(\theta _{a,b}: \langle \langle ba \rangle \rangle  \rightarrow \langle \langle ab \rangle \rangle \)  for all \((a,b)\)  in the set
\(\big \lbrace (\mathrm {U}_0,\mathrm {U}_0), (\mathrm {U}_1,\mathrm {U}_1),(f,g),(g,f),(gf,\mathrm {U}_0),(\mathrm {U}_1,fg), (fg,fg),(gf,gf)\big \rbrace ,\) 
satisfying the conditions in Def:shadow.
Since \({A}\mathrm {dj}_{\le 0}\)  is a strict bicategory, the unitors of \({A}\mathrm {dj}_{\le 0}\)  are identity maps. eq:shadow cond two thus implies that all of the functors
\(\theta _{\mathrm {U}_0,\mathrm {U}_0}: \langle \langle \mathrm {U}_0\mathrm {U}_0 \rangle \rangle  \xrightarrow{} \langle \langle \mathrm {U}_0\mathrm {U}_0 \rangle \rangle \) 

\(\theta _{\mathrm {U}_1,\mathrm {U}_1}: \langle \langle \mathrm {U}_1\mathrm {U}_1 \rangle \rangle  \xrightarrow{} \langle \langle \mathrm {U}_1\mathrm {U}_1 \rangle \rangle \) 

\(\theta _{\mathrm {U}_0,gf}: \langle \langle gf\mathrm {U}_0 \rangle \rangle  \xrightarrow{} \langle \langle \mathrm {U}_0gf \rangle \rangle \) 

\(\theta _{fg,\mathrm {U}_1}: \langle \langle \mathrm {U}_1fg \rangle \rangle  \xrightarrow{} \langle \langle fg\mathrm {U}_1 \rangle \rangle \) 
are just identities. Moreover, eq:shadow cond one implies that the following diagram in \(\begin{center}\begin{tikzcd}(gf)(gf) [dr, "\theta _{gfg,f}"^{\prime }] [rr, "\theta _{gf,gf}"] & & (gf)(gf) \\& f(gf)g [ur, "\theta _{fgf,g}"^{\prime }] &\end{tikzcd}\end{center}commutes, since the associators in \) Adj0\( are identities. Since \) fgf=f\( and \) gfg=g\(, and \) f,g\( and \) g,f\( are mutually inverse, \) gf,gf\( is the identity on \) gf\(. Similarly, we deduce that \) fg,fg\( is the identity on \) fg\(. The only possibly non-identity isomorphisms are therefore \) f,g:gf  fg \( and its inverse \) g,f:fg  gf \(.\) The commutativity of eq:shadow cond one does not impose any conditions on the isomorphism \(\theta _{f,g}\) . Indeed, fix a triple of morphisms \((a:X_0 \rightarrow X_1,b:X_1 \rightarrow X_2,c:X_2 \rightarrow X_0)\)  in the bicategory \({A}\mathrm {dj}_{\le 0}\) . There are two possible cases: either at least one of the three morphisms is an identity map or all three morphisms are non-identities.
If at least two morphisms are the identity, then all \(\theta \)  maps involved are necessarily identities, as \(\theta _{\mathrm {U}_0,\mathrm {U}_0},\theta _{\mathrm {U}_1,\mathrm {U}_1}, \theta _{\mathrm {U}_0,gf},\theta _{fg,\mathrm {U}_1}\)  are all identities.
If only one of \(a,b,\)  and \(c\)  is the identity morphism, then without loss of generality \(a= \mathrm {U}_0\) , in which case the condition of eq:shadow cond one reduces to equality
\(\theta _{b,c} = \theta _{b,c} \circ \mathrm {id}\) 
as the associators are all identities. This condition holds trivially, so there are no additional restrictions on \(\theta _{f,g}\) .
If all three morphisms \(a,b,c\)  are not identities, then there are eight possible scenarios.

\(0 \rightarrow 0 \rightarrow 0 \rightarrow 0: (gf,gf,gf)\)

\(0 \rightarrow 0 \rightarrow 1 \rightarrow 0: (gf,f,g)\)

\(0 \rightarrow 1 \rightarrow 0 \rightarrow 0: (f,g,gf)\)

\(0 \rightarrow 1 \rightarrow 1 \rightarrow 0: (f,fg,g)\)

\(1 \rightarrow 0 \rightarrow 0 \rightarrow 1: (g,gf,f)\)

\(1 \rightarrow 1 \rightarrow 0 \rightarrow 1: (fg,g,f)\)

\(1 \rightarrow 0 \rightarrow 1 \rightarrow 1: (g,f,fg)\)

\(1 \rightarrow 1 \rightarrow 1 \rightarrow 1: (fg,fg,fg)\)

We need to confirm that the commutativity of eq:shadow cond one does not imply any conditions on \(\theta _{f,g}\)  in all 8 scenarios.
By symmetry it suffices to check the first four cases. The first one is trivial as \(\theta _{gf,gf}\)  is the identity map and so \(\theta _{gf,gf} \circ \theta _{gf,gf} = \theta _{gf,gf}\)  holds trivially. For the second case, there is a commutative diagram
(g)(f)(gf) [dr, "gf,gf"'] [rr, "f,g"]   (f)(gf)(g)
 (gf)gf [ur, "f,g"'] 
,
which again commutes trivially as \(\theta _{gf,gf}\)  is the identity. The other two cases are similar, relyin again on the fact that \(\theta _{gf,gf}\)  is the identity. We conclude that specifying a shadow \(\langle \langle - \rangle \rangle \)  from \({A}\mathrm {dj}_{\le 0}\)  to \( is equivalent to providing the data of two functors\)\( \lbrace fg \xrightarrow{} \mathrm {U}_1\rbrace  \rightarrow \quad \lbrace \mathrm {U}_0 \xrightarrow{} gf\rbrace  \rightarrow \)  and an isomorphism \(\theta _{f,g}: \langle \langle gf \rangle \rangle  \rightarrow \langle \langle fg \rangle \rangle \) .
The data required to specify morphisms between such shadows can be specified as follows. Let \(\alpha : \langle \langle - \rangle \rangle _1 \rightarrow \langle \langle - \rangle \rangle _2\)  be a morphism of shadows from \({A}\mathrm {dj}_{\le 0}\)  to \(, i.e., a natural transformation satisfying the condition of {eq:shadow cat cond one}, which implies that the following diagram commutes.\)
U0 1 [r, "u 1"] [d, "U0"]  gf 1 [r, "(1)f,g", ""'] [d, "gf"]  fg 1 [r, "c 1"] [d, "fg"]  U1 1 [d, "U1"]
U0 2 [r, "u 2"]  gf 2 [r, "(2)f,g", ""']  fg 2 [r, "c 2"]  U1 2
Indeed, the naturality of \(\alpha \)  implies that the two outer squares commute, while eq:shadow cat cond one implies the middle square commutes.
There is therefore an equivalence of categories
\({S}\mathrm {ha}({A}\mathrm {dj}_{\le 0}, \simeq \mathrm {Fun}\big ([2]_{Iso},)\) 
that takes a shadow \(\big (\langle \langle - \rangle \rangle ,\theta \big )\)  to the functor \([2]_{Iso} \rightarrow  specified by\)\( \langle \langle \mathrm {U}_0 \rangle \rangle  \xrightarrow{} \langle \langle gf \rangle \rangle  \xrightarrow[\cong ]{\theta _{f,g}} \langle \langle fg \rangle \rangle  \xrightarrow{} \langle \langle \mathrm {U}_1 \rangle \rangle ,\)\((cf. {def:twoiso}).Precomposing this isomorphism with the functor \) Comp:Cocone(Adj0, Sha(Adj0,\( gives an equivalence\)\({U}\mathrm {niv}: {C}\mathrm {ocone}({A}\mathrm {dj}_{\le 0}, \rightarrow \mathrm {Fun}\big ([2]_{Iso},)\)\(that takes a cocone, depicted as a functor\)\(C: \Big \lbrace \begin{tikzcd}[row sep=0.1in, column sep=0.1in]\overset{fg}{\bullet } [d, "c"^{\prime }] \\\underset{\mathrm {U}_1}{\bullet }\end{tikzcd}, \begin{tikzcd}[row sep=0.1in, column sep=0.1in]\overset{\mathrm {U}_0}{\bullet } [d, "u"] \\\underset{gf}{\bullet }\end{tikzcd}\Big \rbrace  \rightarrow \) 
satisfying appropriate compatibility conditions, to the diagram
\( C(\mathrm {U}_0) \xrightarrow{} C(gf) \xrightarrow{} C(fg) \xrightarrow{} C(\mathrm {U}_1).\) 
It follows that \([2]_{Iso}\)  satisfies the universal property of the cocone, whence we have the desired equivalence
\(\mathrm {THH}({A}\mathrm {dj}_{\le 0}) \simeq [2]_{Iso},\) 
given by the colimit diagrams in the statement of the theorem.
We can now use this computation to recover Morita invariance of shadows, as a consequence of the existence of a lift of the Euler characteristic construction to a functor. To formulate this result, we need a two more definitions.
Definition 4.8 
Let \({A}\mathrm {dj}{E}\mathrm {q}\)  be the strict bicategory with the same objects, 1-cells and 2-cells as \({A}\mathrm {dj}\) , but where all 2-morphisms are invertible.

As in rmk:adj picture, \({A}\mathrm {dj}{E}\mathrm {q}\)  can be depicted as a diagram
[row sep=0.5in, column sep=0.5in]
0 [r, bend left = 40, "f"] [loop, in=130, out=225, looseness=10, ""name=D, below, "g f"]  1 [l, bend left =40, "g"] [loop, in=45, out=315, looseness=10, ""name=U, "f g"']
[from=1-1, to=D, Rightarrow, "u" description, ""' near end]
[from=U, to=1-2, Rightarrow, "c" description, ""' near start]
Note there is an evident localization functor \({A}\mathrm {dj}\rightarrow {A}\mathrm {dj}{E}\mathrm {q}\) .
Definition 4.9 
For any bicategory \({B}\) , let
\( {A}\mathrm {dj}({B}) = \mathrm {Fun}({A}\mathrm {dj}, {B})\) 
and
\( {A}\mathrm {dj}{E}\mathrm {q}({B}) = \mathrm {Fun}({A}\mathrm {dj}{E}\mathrm {q},{B}),\) 
the categories of adjunctions in \({B}\)  and of adjoint equivalences in \({B}\) , respectively.

Note that the localization functor \({A}\mathrm {dj}\rightarrow {A}\mathrm {dj}{E}\mathrm {q}\)  induces a restriction functor
\( {A}\mathrm {dj}{E}\mathrm {q}({B}) \rightarrow {A}\mathrm {dj}({B}).\) 
To complete our set-up, we introduce the following notation.
Notation 4.11 For any category \(, let\)\( {A}\mathrm {rr}( = \mathrm {Fun}([1],,\)\(and let \) Iso(\( denote the full subcategory of \) Arr(\( consisting of isomorphisms.\)
We can now prove that the Euler characteristic construction introduced by Campbell and Ponto is actually functorial.
Theorem 4.12 
For any bicategory \({B}\) , there is a functor
\(\widehat{\chi }: {A}\mathrm {dj}({B}) \rightarrow {A}\mathrm {rr}\big (\mathrm {THH}({B})\big )\) 
that takes an adjunction \(\sigma =(C,D,F,G,u,c)\)  to the morphism
\(\widehat{\chi }(\sigma ) : U_C \xrightarrow{} GF \xrightarrow{} FG \xrightarrow{} U_D\) 
in \(\mathrm {THH}({B})\) .

Note that it is essential here that the codomain of this functor is the category of morphisms in \(\mathrm {THH}({B})\) , which ensures the existence of the isomorphism \(GF \xrightarrow{} FG\) .
The functoriality of \(\mathrm {THH}\)  for enrichments in \( ({C}\mathrm {at},\times )\)  (rem:thh bicat functorial) implies that there is a functor
\( \widehat{\mathrm {THH}}: {A}\mathrm {dj}({B}) \rightarrow \mathrm {Fun}\big (\mathrm {THH}({A}\mathrm {dj}),\mathrm {THH}({B})\big ).\) 
We now have following diagram, which factors the diagram in thm:thh adj

[column sep=0.2in, row sep=0.5in]
[0 ][0 ][d, "(f,id,g) (f,g ,id)"]
[r] [r, shift left=2] [r, shift right=2]

[0 ][d, "(f ,g)"]
[r, shift left=1.2, "1"] [r, shift right=1.2, "1'"']
[l, shift right, shorten >=1ex,shorten <=1ex ] [l, shift left, shorten >=1ex,shorten <=1ex]

{0 1 } {1' 2} [d, "u c"]
[l, shorten >=1ex,shorten <=1ex] [r]

[2 ]Iso [d] [dd, bend left=45, ""]
X0,X1,X2 Adj(X0,X1,X2,X0) [d, "0"]
[r] [r, shift left=2, "d2"] [r, shift right=2, "d0"']

X0,X1 Adj(X0,X1,X0) [d, "0"]
[r, shift left=1.2, "d1"] [r, shift right=1.2, "d0"']
[l, shift right, shorten >=1ex,shorten <=1ex ] [l, shift left, shorten >=1ex,shorten <=1ex]

Adj(0,0) Adj(1,1) [d, "0"]
[l, shorten >=1ex,shorten <=1ex] [r]

THH(Adj) [d, "THH(0)"]
X0,X1,X2 Adj0(X0,X1,X2,X0)
[r] [r, shift left=2, "d2"] [r, shift right=2, "d0"']

X0,X1 Adj0(X0,X1,X0)
[r, shift left=1.2, "d1"] [r, shift right=1.2, "d0"']
[l, shift right, shorten >=1ex,shorten <=1ex ] [l, shift left, shorten >=1ex,shorten <=1ex]

Adj0(0,0) Adj0(1,1)
[l, shorten >=1ex,shorten <=1ex] [r]

THH(Adj0)
.
The composition of the top to bottom is the isomorphism given in thm:thh adj and so we can deduce that the map \(\mathrm {THH}(\pi _{\le 0})\)  has a section \([2]_{Iso} \rightarrow \mathrm {THH}({A}\mathrm {dj})\) . We can precompose REF  with this section to get a map
\(\mathrm {Fun}\big (\mathrm {THH}({A}\mathrm {dj}),\mathrm {THH}({B})\big ) \rightarrow \mathrm {Fun}\big ([2]_{Iso},\mathrm {THH}({B})\big ).\) 
which takes an adjunction \(\sigma =(C,D,F,G,u,c)\)  to the sequence of composable morphisms \(\mathrm {U}_C \xrightarrow{} FG \xrightarrow{} GF \xrightarrow{} \mathrm {U}_D\) .
We now define \(\widehat{\chi }\)  to be the composite functor
\({A}\mathrm {dj}({B}) \xrightarrow{} \mathrm {Fun}\big ([2]_{Iso},\mathrm {THH}({B})\big ) \rightarrow \mathrm {Fun}\big ([1],\mathrm {THH}({B})\big ) = {A}\mathrm {rr}(\mathrm {THH}({B}))\) 
where the second functor composes the morphisms in a sequence.
By construction \(\widehat{\chi }\)  takes an adjunction \(\sigma =(C,D,F,G,u,c)\)  to the desired morphism \(U_C \rightarrow U_D\) .
Remark 4.15 The key to the proof above is the computation of \(\mathrm {THH}({A}\mathrm {dj}_{\le 2})\)  (thm:thh adj). We hope that in the future, similar computations can help us better understand further invariance properties of \(\mathrm {THH}\)  and shadows.
Moreover, thm:chi functor will generalize to an arbitrary \(-enriched \)\(-category, as soon as the functoriality of \) THH\( is established in this case.\)
Remark 4.16 Our proof relied on the fact that the quotient
\(\mathrm {THH}(\pi _{\le 0}): \mathrm {THH}({A}\mathrm {dj}) \rightarrow \mathrm {THH}({A}\mathrm {dj}_{\le 0})\) 
admits a section. It should be noted this section does not come from a functor \({A}\mathrm {dj}_{\le 0} \rightarrow {A}\mathrm {dj}\) .

Remark 4.17 We did not need to compute \(\mathrm {THH}({A}\mathrm {dj})\)  itself, but rather an approximation thereof, which sufficed to deduce Morita invariance. In recent work Ayala and Francis have (independently) computed \(\mathrm {THH}({A}\mathrm {dj})\)  [32]} using methods from factorization homology [33]}.

The corollary below is an immediate consequence of the fact that every functor preserves isomorphisms.
Corollary 4.18 
Precomposing the functor \( \widehat{\chi }: {A}\mathrm {dj}({B}) \rightarrow {A}\mathrm {rr}\big (\mathrm {THH}({B})\big )\)  with the restriction functor \({A}\mathrm {dj}{E}\mathrm {q}({B}) \rightarrow {A}\mathrm {dj}({B})\)  given in REF  gives rise to a functor
\(\widehat{\chi }: {A}\mathrm {dj}{E}\mathrm {q}({B}) \rightarrow {I}\mathrm {so}\big (\mathrm {THH}({B})\big ).\) 


From cor:chi functor on iso we can recover the Morita invariance of shadows, proven originally in [1]}.
Proposition 4.19 Shadows satisfy Morita invariance.

Let \(\langle \langle - \rangle \rangle \)  be a shadow functor on \({B}\) . Consider its unstrictification
(thm:main theorem), which is a functor \({U}\mathrm {n}\big (\langle \langle - \rangle \rangle \big ): \mathrm {THH}({B}) \rightarrow . Precomposing this with the functor \)\( of {thm:chi functor} gives rise to a functor\)\( {A}\mathrm {rr}\Big ({U}\mathrm {n}\big (\langle \langle - \rangle \rangle \big )\Big ) \circ \widehat{\chi }: {A}\mathrm {dj}({B}) \rightarrow {A}\mathrm {rr}(\)\(that takes an adjunction in \) B\( to the morphism\)\(\langle \langle U_C \rangle \rangle  \rightarrow \langle \langle GF \rangle \rangle  \cong \langle \langle FG \rangle \rangle  \rightarrow \langle \langle U_D \rangle \rangle \)\(in \) .
The desired conclusion now follows from cor:chi functor on iso.
Remark 4.20 The Euler characteristic construction of Campbell and Ponto is a special case of an important construction that they call the trace of a 2-cell [1]}. It could be interesting to construct a functorial lift of the trace construction as well, which would require computing THH of a extension of \({A}\mathrm {dj}\)  to a more complex strict bicategory.

Towards traces in \((\infty ,2)\) -category theory
We are finally in a position to generalize shadows to \((\infty ,2)\) -categories. Motivated by thm:main theorem, we define a shadow on an \((\infty ,2)\) -category \({B}\)  with values in an \((\infty ,1)\) -category \( to be a trace ({defn:trace}), i.e., a functor\)\(\mathrm {THH}({B}) \rightarrow \)\(where \) THH(B)\( is constructed as in {def:thh of enriched cat}.\) We should think of this definition as a homotopy coherent lift of the definition of a shadow. To formulate this concretely, we need the following definition.
Definition 5.1 Let \({B}\)  be an \((\infty ,2)\) -category specified by a symmetric monoidal functor
\({B}: {B}\mathrm {ypass}_S \rightarrow {C}\mathrm {at}_\infty .\) 
Its homotopy bicategory \(\mathrm {Ho}({B})\)  is the composite
\(\mathrm {Ho}\circ {B}: {B}\mathrm {ypass}_S \rightarrow {C}\mathrm {at}_\infty \rightarrow {C}\mathrm {at}.\) 


Note that the definition does indeed give rise to a bicategory (i.e. enriched over \({C}\mathrm {at}\)  in the sense of def:enriched infty cat) because \(\mathrm {Ho}\)  preserves finite products and thus is symmetric monoidal.
Proposition 5.2 Let \({B}\)  be an \((\infty ,2)\) -category and \( an \) (,1)\(-category. There is an equivalence\)\({S}\mathrm {ha}\big (\mathrm {Ho}({B}),\mathrm {Ho}(\big ) \simeq \mathrm {Fun}\big (\mathrm {THH}(\mathrm {Ho}({B})),\mathrm {Ho}(\big ) \simeq \mathrm {Fun}\big (\mathrm {THH}({B}),\mathrm {Ho}(\big ).\)\(\)
The first equivalence is an immediate consequence of thm:main theorem.
The second follows from the fact that \(\mathrm {Ho}\)  commutes with colimits and so
\(\mathrm {THH}(\mathrm {Ho}({B})) \simeq \mathrm {Ho}(\mathrm {THH}({B})),\) 
combined with the fact that \(\mathrm {Ho}\)  is a left adjoint (subsec:notation).
In particular, there is a localization functor
\(\mathrm {Fun}(\mathrm {THH}({B}),  \rightarrow {S}\mathrm {ha}(\mathrm {Ho}{B},\mathrm {Ho}\) 
that takes a trace of \((\infty ,2)\) -categories to a shadow. This observation leads to the following natural question.
Question 5.3 Let \({B}\)  be an \((\infty ,2)\) -category and
\(T: \mathrm {THH}(\mathrm {Ho}{B}) \rightarrow \mathrm {Ho}\) 
a shadow on \(\mathrm {Ho}{B}\) . Can the shadow be lifted to a trace that makes the following diagram commute?
[row sep=0.5in, column sep=0.5in]
THH(B) [r, dashed] [d]  [d]
THH(HoB) [r, "T"]  Hotikzcd

A general answer to this question might be negative, but there are certain cases where we do expect this to hold.
Example 5.4 Campbell and Ponto's work [1]} implies that topological Hochschild homology of a spectral category extends to a trace functor from the bicategory of bimodules over spectral categories to the homotopy category of spectra
\(\mathrm {THH}\big (\mathrm {Ho}({M}\mathrm {od})\big ) \rightarrow \mathrm {Ho}(\mathrm {Sp}).\) 
See also rmk:thh trace.
We expect this particular trace to lift to an \((\infty ,1)\) -functor
\(\mathrm {THH}({M}\mathrm {od}) \rightarrow \mathrm {Sp}.\) 


appendix

Truncated Simplicial Pseudo-Diagrams
In this appendix we perform a detailed analysis of pseudofunctors out of the truncated simplex category \((\Delta _{\le 2})^{op}\) , which is essential for our proof of thm:main theorem.
A 2-category is by definition a category enriched over categories. Hence, there is a default notion of morphisms between 2-categories, called 2-functors, which are strict functors that respect the categorical enrichment. On the other hand, we can also consider pseudofunctors, for which the composition and unit hold only up to chosen natural isomorphisms. For a more detailed discussion of pseudofunctors see [37]}, [38]}.
We want to characterize pseudofunctors out of \((\Delta _{\le 2})^{op}\)  and out of \(((\Delta _{\le 2})^{op})^{\rhd }\)  into a 2-category \({B}\) . Constructing a pseudofunctor by hand can be quite challenging, however, since we need to specify a natural isomorphism for every pair of composable morphisms in the domain. Fortunately, there is a way to simplify the task. In [39]}, [40]} Lack constructs a model structure on the category of 2-categories that has the property that any pseudofunctor between 2-categories with cofibrant domain is equivalent to a 2-functor [39]}.
We start therefore by recalling the characterization of the cofibrant objects in Lack's model structure, which requires the notion of free categories. Recall that a graph is a specified by a pair of functions with the same domain and the same codomain. Every small category has an underlying graph, given by the source and target functions on the set of morphisms, i.e., there is a forgetful functor \(U:{C}\mathrm {at}\rightarrow {G}\mathrm {rph}\) . The forgetful functor admits a left adjoint \({F}_1:{G}\mathrm {rph}\rightarrow {C}\mathrm {at}\) , which takes a graph to its free category. For more details regarding free categories and graphs see [23]}. The following result regarding cofibrant objects summarizes [39]}.
Theorem 1.1 
A 2-category is cofibrant in Lack's model structure if and only its underlying 1-category is a free category on a graph.

Unfortunately, the categories \((\Delta _{\le 2})^{op}\)  and \(((\Delta _{\le 2})^{op})^{\rhd }\)  are not cofibrant in Lack's model structure on 2-categories. Indeed the generating morphisms satisfy non-trivial relations (cf. lemma:trunc simplex gen). Hence our first task is to find appropriate cofibrant replacements, for which we apply the theory of computads introduced in [44]}. A computad consists of a graph \({G}\)  together with a set of 2-arrows between parallel morphisms in the free category on \({G}\) . We refer to the original source [44]} for a more complete description, as we need only certain computads.
Before we proceed, it is helpful to describe fully the category \((\Delta _{\le 2})^{op}\) , in particular its morphisms. As it is a full subcategory of \(\Delta ^{op}\) , we can state the following result in terms of the characterization of \(\Delta \)  via generators and relations found in [46]} and [23]}
Lemma 1.2 
The category \((\Delta _{\le 2})^{op}\)  can be described as follows.

It has three objects: \(0,1,2\) .

It has eight generating morphisms.

\(s_0: 0 \rightarrow 1\)

\(d_0,d_1: 1 \rightarrow 0\)

\(s_0,s_1: 1 \rightarrow 2\)

\(d_0,d_1,d_2: 2 \rightarrow 1\)


The generating morphisms satisfy the following relations.

\(d_0d_0= d_0d_1,d_0d_2=d_1d_0, d_1d_1 =d_1d_2\)

\(d_1s_0=\mathrm {id} =d_0s_0\)

\(d_0s_0=d_1s_1 =\mathrm {id}= d_1s_0=d_2s_1\)

\(s_0s_0=s_1s_0\)



Motivated by this description, we now construct the desired computad, starting with its underlying graph.
Definition 1.3 
Let \(_2\)  be the graph with three objects \(\lbrace 0,1,2\rbrace \)  and eight arrows \(\lbrace s_0:0 \rightarrow 1; d_0,d_1: 1 \rightarrow 0; s_0,s_1: 1 \rightarrow 2; d_0,d_1,d_2: 2 \rightarrow 1\rbrace \) .

Next we add the relevant 2-arrows based on the relations in lemma:trunc simplex gen.
Definition 1.4 
Let \(\mathcal {G}_2\)  be the computad with underlying graph \(_2\) , equipped with following 2-arrows in \({F}_1(_2)\) .

\(d_0d_0\Rightarrow d_0d_1,d_0d_2\Rightarrow d_1d_0, d_1d_1 \Rightarrow d_1d_2\)

\(d_1s_0\Rightarrow \mathrm {id} \Leftarrow d_0s_0\)

\(d_0s_0\Rightarrow d_1s_1 \Rightarrow \mathrm {id}\Leftarrow d_1s_0 \Leftarrow d_2s_1\)

\(s_0s_0\Rightarrow s_1s_0\)


Applying [44]} to the computad \(\mathcal {G}_2\)  gives us following result.
Lemma 1.5 
There is a 2-category \({F}_2(\mathcal {G}_2)\)  satisfying the universal property that every 2-functor \({F}_2(\mathcal {G}_2) \rightarrow {B}\)  is specified by the following data.

A functor from the free category \({F}_1(_2)\)  on the graph of \(_2\)  to the underlying category of \({B}\) .

A choice of 2-morphism in \({B}\)  with the appropriate source and target for every 2-arrow of the computad \(\mathcal {G}_2\) .


We refer to [44]} for a detailed explanation of what “appropriate" means in this context.
Applying the universal property of the free category functor, let
\(\pi _1:{F}_1(_2) \rightarrow (\Delta _{\le 2})^{op}\) 
denote the functor that is the identity on objects and generating 1-morphisms. Note that by the universal property of \({F}_2(\mathcal {G}_2)\) , there is a 2-functor
\(\pi _2:{F}_2(_2) \rightarrow (\Delta _{\le 2})^{op}\) 
with underlying functor \(\pi _1\)  and sending every 2-arrow of the computad \(_2\)  to an identity.
Before proceeding further, we analyze the structure of the 2-category \({F}_2(\mathcal {G}_2)\)  and the relations satisfied by its 2-morphisms.
Lemma 1.6 
The 2-functor \(\pi _2:{F}_2(\mathcal {G}_2) \rightarrow (\Delta _{\le 2})^{op}\)  is a cofibrant replacement in Lack's model structure on bicategories.

By thm:lack cofibrant, the 2-category \({F}_2(\mathcal {G}_2)\)  is evidently cofibrant as its underlying category is just \({F}_1(_2)\) , which is free by definition.
We therefore need only to show that the projection map
\(\pi _2:{F}_2(_2) \rightarrow (\Delta _{\le 2})^{op}\) 
is a weak equivalence in Lack's model structure. However, as follows from the proof of [39]} and the explanation immediately thereafter, this functor is actually a trivial fibration.
We apply this result to establish a useful characterization of the 2-category \({F}_2(\mathcal {G}_2)\) . To do so, it is helpful to review contractible groupoids.
Lemma 1.7 
The forgetful functor \({G}\mathrm {rpd}\rightarrow {S}\text{et}\) , which takes a groupoid to its set of objects, admits a right adjoint \(I: {S}\text{et}\rightarrow {G}\mathrm {rpd}\) , which takes a set \(S\)  to the groupoid \(I(S)\)  with the same set of objects and a unique isomorphism between any two objects. Moreover, a category with object set \(S\)  is contractible if and only if it is isomorphic to \(I(S)\) .

Only the last sentence requires an argument. If \( is a contractible category with object set \) S\(, then for any object \) s S\(, the functor \) s:[0]  is an equivalence and thus fully faithful, which implies that \(s,s)= \lbrace \mathrm {id}_s\rbrace \) .
Let \(s,s^{\prime } \in S\) . Since the map \(s: [0] \rightarrow  is an equivalence, it is essentially surjective, whence there exists an isomorphism \) s s'\( in \) , which must be unique. Indeed, the existence of two distinct isomorphisms \(f,g: s \rightarrow s^{\prime }\)  would imply the existence of a non-trivial automorphism \(g^{-1}f:s \rightarrow s\) , in contradiction with the conclusion of the previous paragraph. We conclude that \( is isomorphic to \) I(S)\(, as desired.\)
We call \(I(S)\)  the contractible groupoid based on the set \(S\), as it is equivalent to the terminal category. We can formulate an alternative characterization of the morphism categories in \({F}_2(\mathcal {G}_2)\)  in terms of contractible groupoids.
Lemma 1.8 
The 2-category \({F}_2(\mathcal {G}_2)\)  has three objects \(0,1,2\) . Moreover, for any \(i,j\in \lbrace 0,1,2\rbrace \) , the category \({F}_2(\mathcal {G}_2)(i,j)\)  has as objects the set \({F}_1(_2)(i,j)\) , while for two objects \(f,g\) , there is a unique morphism from \(f\)  to \(g\)  if and only and only if \(\pi _1(f) = \pi _1(g)\)  in \((\Delta _{\le 2})^{op}\) . In particular, \({F}_2(\mathcal {G}_2)(i,j)\)  is a groupoid, and there is an isomorphism of groupoids
\({F}_2(\mathcal {G}_2)(i,j) \cong \coprod _{f \in (\Delta _{\le 2})^{op}(i,j)} I\big ((\pi _1)^{-1}(f)\big ).\) 


The characterization of the underlying category of the 2-category \({F}_2(\mathcal {G}_2)\)  follows from lemma:cg computad inverse. For all \(i,j \in \lbrace 0,1,2\rbrace \) , lemma:cofib rep implies that
\({F}_2(\mathcal {G}_2)(i,j) \rightarrow (\Delta _{\le 2})^{op}(i,j)\) 
is an equivalence of categories. Since the right hand side is discrete, \({F}_2(\mathcal {G}_2)(i,j)\)  is a disjoint union of contractible categories. The desired isomorphism now follows from the characterization of contractible categories given in lemma:contractible groupoid.
We next use this cofibrant replacement to characterize pseudofunctors out of \((\Delta _{\le 2})^{op}\) . To simplify notation, we henceforth denote the cofibrant replacement above by
\(\pi _Q: Q(\Delta _{\le 2})^{op} \rightarrow (\Delta _{\le 2})^{op}.\) 
The next result is an immediate consequence of lemma:alt description ftwogtwo and [39]}.
Corollary 1.9 
Let \({B}\)  be a 2-category.
Precomposing with the cofibrant replacement functor \(\pi _Q:Q(\Delta _{\le 2})^{op} \rightarrow (\Delta _{\le 2})^{op}\)  induces an equivalence between the category of pseudofunctors \((\Delta _{\le 2})^{op} \rightarrow {B}\)  and that of 2-functors \(Q(\Delta _{\le 2})^{op} \rightarrow {B}\) . In other words, for every pseudofunctor \(F: (\Delta _{\le 2})^{op} \rightarrow {B}\) , there exists a 2-functor \(\widehat{F}:Q(\Delta _{\le 2})^{op} \rightarrow {B}\)  and an equivalence \(F \circ \pi _Q \simeq \widehat{F}\) .

Combining the results above, we can now characterize pseudofunctors out of \((\Delta _{\le 2})^{op}\) .
Lemma 1.10 
For any 2-category \({B}\) , a pseudofunctor \((\Delta _{\le 2})^{op} \rightarrow {B}\)  is determined up to equivalence by a choice of the following data.

Three objects \(B_0,B_1,B_2\)

Eight 1-morphisms

\(s_0: B_0 \rightarrow B_1\)

\(d_0,d_1: B_1 \rightarrow B_0\)

\(s_0,s_1: B_1 \rightarrow B_2\)

\(d_0,d_1,d_2: B_2 \rightarrow B_1\)


Ten invertible 2-morphisms

\(d_0d_0 \overset{\alpha _0}{\Rightarrow } d_0d_1,d_0d_2\overset{\alpha _1}{\Rightarrow } d_1d_0, d_1d_1 \overset{\alpha _2}{\Rightarrow } d_1d_2\)  in \({B}(B_2,B_0)\)

\(d_1s_0 \overset{\beta _0}{\Rightarrow } \mathrm {id} \overset{\beta _1}{\Leftarrow } d_0s_0\)  in \({B}(B_0,B_0)\)

\(d_0s_0 \overset{\gamma _0}{\Rightarrow } d_1s_1 \overset{\gamma _1}{\Rightarrow } \mathrm {id} \overset{\gamma _2}{\Leftarrow } d_1s_0 \overset{\gamma _3}{\Leftarrow } d_2s_1\)  in \({B}(B_1,B_1)\)

\(s_0s_0 \overset{\delta _0}{\Rightarrow } s_1s_0\)  in \({B}(B_0,B_2)\)


Moreover, any two 2-morphisms generated by the 2-morphisms above with equal domain and codomain are equal.

By cor:pseudo vs two a pseudofunctor \((\Delta _{\le 2})^{op} \rightarrow {B}\)  is determined up to equivalence by a 2-functor \(Q(\Delta _{\le 2})^{op} \rightarrow {B}\) . The universal property of \(Q(\Delta _{\le 2})^{op}\)  as formulated in lemma:cg computad inverse implies that a 2-functor \(Q(\Delta _{\le 2})^{op} \rightarrow {B}\)  is specified by a choice of \(0,1,2\) -morphisms and relations as above.
Now that we have a useful description of pseudofunctors out of \((\Delta _{\le 2})^{op}\) , the next step is to study pseudofunctorial lifts to \(((\Delta _{\le 2})^{op})^\rhd \) . Recall that the category \(((\Delta _{\le 2})^{op})^\rhd \)  is the join of \((\Delta _{\le 2})^{op}\)  and a final object (def:cocone). To characterize pseudofunctors out of \(((\Delta _{\le 2})^{op})^{\rhd }\)  directly is challenging, so we prefer instead to study 2-functors out of its cofibrant replacement, which requires proving results analogous to lemma:cg computad inverse and lemma:trunc simp diag. First we describe \(((\Delta _{\le 2})^{op})^\rhd \)  in terms generators and relations, analogously to lemma:trunc simplex gen.
Lemma 1.11 
Let \(((\Delta _{\le 2})^{op})^\rhd \)  be the category constructed by joining a terminal object to \((\Delta _{\le 2})^{op}\) . More precisely, \(((\Delta _{\le 2})^{op})^\rhd \)  is specified by following generators and relations.

Objects \(0,1,2,f\)

Morphisms

The generating morphisms in \((\Delta _{\le 2})^{op}\) : \(s_0: 0 \rightarrow 1\) , \(d_0,d_1: 1 \rightarrow 0\) , \(s_0,s_1: 1 \rightarrow 2\) , \(d_0,d_1,d_2: 2 \rightarrow 1\) ,

Three additional morphisms: \(t_0:0 \rightarrow f\) , \(t_1:1 \rightarrow f\) , \(t_2: 2 \rightarrow f\) .


Relations

The relations that already hold in \((\Delta _{\le 2})^{op}\) :

\(d_0d_0=d_0d_1\) , \(d_0d_2= d_1d_0\) , \(d_1d_1= d_1d_2: 2 \rightarrow 0\) ,

\(d_1s_0 = \mathrm {id} = d_0s_0:0 \rightarrow 0\) ,

\(d_0s_0 =d_1s_1= \mathrm {id}=d_1s_0=d_2s_1:1 \rightarrow 1\) ,

\(s_0s_0=s_1s_0:0 \rightarrow 2\) ,


The additional relations for the terminal object:

\(t_0=t_1s_0=t_2s_0s_0=t_2s_1s_0:0 \rightarrow f\) ,

\(t_1=t_0d_0=t_0d_1=t_2s_0=t_2s_1: 1 \rightarrow f\) ,

\(t_2=t_1d_0=t_1d_1=t_1d_2=t_0d_0d_0=t_0d_0d_1=t_0d_0d_2= t_0d_1d_0=t_0d_1d_2= t_0d_1d_1:2 \rightarrow f\) .




There is an obvious fully faithful inclusion \((\Delta _{\le 2})^{op} \rightarrow ((\Delta _{\le 2})^{op})^{\rhd }\) . The list of relations for \(((\Delta _{\le 2})^{op})^{\rhd }\)  given above contains many redundancies, which we can reduce as follows.
Lemma 1.12 
The category \(((\Delta _{\le 2})^{op})^{\rhd }\)  can be specified by the following data.

The same objects as above.

The generating morphisms in \((\Delta _{\le 2})^{op}\)  together with one additional morphism \(t_0:0 \rightarrow f\)

The relations that hold in \((\Delta _{\le 2})^{op}\)  together with one additional relation \(t_0d_0=t_0d_1: 1 \rightarrow f\) .


We define \(t_1=t_0d_0\)  and \(t_2=t_0d_0d_0\) .
Since the relations listed in this lemma are a subset of the relations in def:deltaleqtwofinal, the necessity is evident. We prove that these relations suffice by recovering the remaining relations from them, as follows.

\(t_2s_0s_0 \overset{s_0s_0=s_1s_0}{=}t_2s_1s_0\) .

\(t_2\overset{def}{=}t_0d_0d_0\overset{t_1=t_0d_0}{=}t_1d_0\)

\(t_2\overset{def}{=} t_0d_0d_0\overset{d_0d_0=d_0d_1}{=}t_0d_0d_1\overset{t_0d_0=t_1}{=}t_1d_1\)

\(t_2\overset{def}{=}t_0d_0d_0\overset{t_1=t_0d_0}{=}t_1d_0\overset{t_1=t_0d_1}{=}t_0d_1d_0\overset{d_1d_0=d_0d_2}{=}t_0d_0d_2\overset{t_0d_0=t_1}{=}t_1d_2\)

\(t_2 \overset{def}{=} t_0d_0d_0\overset{t_0d_0=t_0d_1}{=} t_0d_1d_0\overset{d_1d_0=d_0d_2}{=}t_0d_0d_2\overset{t_0d_0=t_0d_1}{=}t_0d_1d_2\overset{d_1d_2=d_1d_1}{=}t_0d_1d_1\)

\(t_2s_0\overset{t_2=t_0d_0d_0}{=}t_0d_0d_0s_0\overset{t_1=t_0d_0}{=}t_1d_0s_0 \overset{d_0s_0=\mathrm {id}}{=} t_1\)

\(t_2s_1\overset{t_2=t_0d_0d_0}{=} t_0d_0d_0s_1\overset{d_0d_0=d_0d_1}{=}t_0d_0d_1s_1\overset{t_0d_0=t_1}{=}t_1d_1s_1\overset{d_1s_1=\mathrm {id}}{=}t_1\)

\(t_2s_0s_0\overset{t_2=t_0d_0d_0}{=}t_0d_0d_0s_0s_0\overset{\mathrm {id} = d_0s_0}{=}t_0d_0s_0 \overset{\mathrm {id} = d_0s_0}{=}t_0\)

\(t_1s_0\overset{t_1=t_0d_0}{=}t_0d_0s_0 \overset{d_0s_0=\mathrm {id}}{=} t_0\)

As we have recovered all the equalities in def:deltaleqtwofinal, the result follows.
Given this lemma, we can easily modify the proofs of lemma:cg computad inverse and lemma:trunc simp diag to establish the following analogous result.
Definition 1.13 
Let \((_2)^{\rhd }\)  denote the graph with

vertices \(0,1,2,f\) , and

the same edges as in \(_2\)  (def:cdtwo), together with one additional edge \(0 \rightarrow f\) .


Lemma 1.14 
The 2-category \(\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\)  admits a cofibrant replacement given by the 2-category \(Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big )\) , which is the 2-category determined by the computad with

underlying graph is \((_2)^\rhd \) , and

the same 2-arrows as \(\mathcal {G}_2\)  (def:cg computad), along with one additional 2-arrow \(\theta : t_0d_0 \Rightarrow t_0d_1\) .


The morphism categories of \(Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big )\)  can be characterized as in lemma:alt description ftwogtwo, leading in particular to the following observation.
Corollary 1.15 
The category \(Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big )(i,f)\)  is a contractible groupoid for all \(i\in \lbrace 0,1,2\rbrace \) .

The proof is essentially the same as that of lemma:alt description ftwogtwo, given that there are unique maps \(i \rightarrow f\)  in \(Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big )\) .
Remark 1.16 
cor:contractible-join enables us to better understand \(Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big )\) . There are four objects \(0,1,2,f\) , and morphisms are words with letters
\(s_0,d_0,d_1,s_0,s_1,d_0,d_1,d_2,t_0.\) 
In particular, the only generating morphism that has codomain \(f\)  is \(t_0:0 \rightarrow f\) , whence every object in the groupoid \(Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big )(i,f)\)  is necessarily of the form \(t_0 g\)  for some morphism \(g: i \rightarrow 0\) , where \(i \in \lbrace  0,1,2\rbrace \) . Contractibility of \(Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big )(i,f)\)  implies that there is precisely one isomorphism between \(t_0g_0\)  and \(t_0g_1\)  for any \(g_0,g_1: i \rightarrow 0\) .

The equivalence below of 2-functors and pseudofunctors follows from lemma:cofib rep final and [39]} .
Corollary 1.17 
Let \({B}\)  be a 2-category.
Precomposing with the cofibrant replacement functor
\(\pi _Q^\rhd :Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big ) \rightarrow \big ((\Delta _{\le 2})^{op}\big )^{\rhd }\) 
induces an equivalence between the category of pseudofunctors \(\big ((\Delta _{\le 2})^{op}\big )^{\rhd } \rightarrow {B}\)  and that of 2-functors \(Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big ) \rightarrow {B}\) . In other words, for every pseudofunctor \(F: \big ((\Delta _{\le 2})^{op}\big )^{\rhd } \rightarrow {B}\) , there exists a 2-functor \(\widehat{F}:Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big ) \rightarrow {B}\)  and an equivalence \(F \circ \pi _Q^\rhd \simeq \widehat{F}\) .

We can use this result to characterize pseudofunctors out of \(((\Delta _{\le 2})^{op})^{\rhd }\) , using the inclusion functor \(Q(\Delta _{\le 2})^{op} \rightarrow Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big )\)  arising from the inclusion of computads. In the statement and the proof below, we use the notation of lemma:trunc simp diag.
Lemma 1.18 
Let \({B}\)  be a 2-category, and let \(F:(\Delta _{\le 2})^{op} \rightarrow {B}\)  be a pseudofunctor. An extension of \(F\)  to \(\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\)  is specified by
(2)op [d, hookrightarrow] [r, "F"]  B
((2)op) [ur, "F"', dashed]

an object \(B_f\)  in \({B}\) ,

a 1-morphism \(t_0: B_0 \rightarrow B_f\) , and

an invertible 2-morphism \(t_0d_0 \overset{\theta }{\Rightarrow } t_0d_1\)

such that the following equalities hold, where \(*\)  denotes whiskering of 2-cells by 1-cells, and \(\circ \)  denotes vertical composition of 2-cells.

\(\theta * s_0= t_0*((\beta _0)^{-1} \circ \beta _1): t_0d_0s_0 \rightarrow t_0d_1s_0\)

\( (t_0 (\alpha _1)^{-1}) \circ (\theta * d_0) = (\theta ^{-1} * d_2) \circ (t_0 * \alpha _2) \circ (\theta * d_1) \circ (t_0 * \alpha _0):t_0d_0d_0 \rightarrow t_0d_0d_2\)


By cor:pseudo vs two and cor:ps-2, we can instead start with a 2-functor \(F:Q((\Delta _{\le 2})^{op}) \rightarrow {B}\)  and define a lift \(\widehat{F}: Q((\Delta _{\le 2})^{op})^{\rhd } \rightarrow {B}\) . The nature of the data required to specify the image of \(0,1\) ,and 2-morphisms under \(\widehat{F}\)  follows from the characterization of \(Q((\Delta _{\le 2})^{op})^{\rhd }\)  in lemma:cofib rep final.
In order to complete the proof we make explicit the relations that guarantee that the groupoids \(Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big )(i,f)\)  are in fact contractible, i.e., that for any two morphisms \(g_1,g_2: i \rightarrow f\) , there is a unique natural isomorphism \(g_1 \cong g_2\) . Any other natural isomorphism generated by \(\theta :t_0d_0\cong t_0d_1\)  must therefore coincide with the original one.
We start by analyzing \(Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big )(1,f)\) . As explained in rem:contractible-maps, every object in this category is of the form \(t_0g^{\prime }\)  for some \(g^{\prime }:1 \rightarrow 0\) . By construction, every object \(g^{\prime }\)  in \(Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big )(1,0)\)  is uniquely  isomorphic to either \(d_0:1 \rightarrow 0\)  or \(d_1:1 \rightarrow 0\) . For any two 1-cells \(g_1,g_2:1 \rightarrow t\) , with \(g_1=t_0g_1^{\prime }\) , \(g_2=t_0g_2^{\prime }\) , there are thus four possible scenarios.

There are unique isomorphisms \(g_1^{\prime } \overset{\kappa _1}{\cong } d_0\)  and \(g_2^{\prime } \overset{\kappa _2}{\cong } d_0\) , whence
\(g_1= t_0g_1^{\prime } \overset{t_0\kappa _1}{\cong } t_0d_0 \overset{t_0\kappa _2}{\cong } t_0g_2^{\prime } = g_2\) 
is the unique isomorphism between \(g_1\)  and \(g_2\) .

There are unique isomorphisms \(g_1^{\prime } \overset{\kappa _1}{\cong } d_1\)  and \(g_2^{\prime } \overset{\kappa _2}{\cong } d_1\) . This case is essentially identical to the previous one.

There are unique isomorphisms \(g_1^{\prime } \overset{\kappa _1}{\cong } d_0\)  and \(g_2^{\prime } \overset{\kappa _2}{\cong } d_1\) . In this case the unique isomorphism between \(g_1\)  and \(g_2\)  is obtained as follows.
\(g_1= t_0g_1^{\prime } \overset{t_0\kappa _1}{\cong } t_0d_0 \overset{\theta }{\cong } t_0d_1 \overset{t_0\kappa _2}{\cong } t_0g_2^{\prime } = g_2\) 


There are unique isomorphisms \(g_1^{\prime } \overset{\kappa _1}{\cong } d_1\)  and \(g_2^{\prime } \overset{\kappa _2}{\cong } d_0\) . This case is the same as the previous one, up to permutation of the roles of \(g_1\)  and \(g_2\) .

Consider now the case of \(Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big )(0,f)\) .
There are three types of objects in this category.

\(t_0g\) , where \(g:0 \rightarrow 0\)  is uniquely isomorphic to the identity in \(Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big )(0,0)\) .

\(t_0d_0g\) , where \(g:0 \rightarrow 1\)  is uniquely isomorphic to \(s_0\)  in \(Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big )(0,1)\) .

\(t_0d_1g\) , where \(g:0 \rightarrow 1\)  is uniquely isomorphic to \(s_0\)  in \(Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big )(0,1)\) .

As in the previous case, there is a unique isomorphism \(t_0d_0g \cong t_0d_1g\) . We need therefore only to compare a 1-cell of the form \(t_0g_1\) , where \(g_1 \cong \mathrm {id}_0\) , and a 1-cell of the form \(t_0d_0g_2\) , where \(g_2 \cong s_0\) , and to find the necessary and sufficient conditions to guarantee that there is a unique isomorphism between \(t_0g_1\)  and \(t_0d_0g_2\) .
By assumption, \(t_0g_1 \cong t_0\)  and \(t_0d_0g_2 \cong t_0d_0s_0\) , via unique isomorphisms. There are two potential isomorphisms between \(t_0\)  and \(t_0d_0s_0\) :

\(t_0d_0s_0 \overset{t_0*\beta _1}{\Rightarrow } t_0\) , and

\(t_0d_0s_0 \overset{\theta * s_0}{\Rightarrow }t_0d_1s_0 \overset{t_0*\beta _0}{\Rightarrow } t_0\) .

Hence, there is a unique isomorphism \(t_0g_1 \cong t_0d_0g_2\)  if and only if \(\theta * s_0 = t_0*((\beta _0)^{-1} \circ \beta _1)\) .
The final case is that of \(Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big )(2,f)\) . If \(g:2 \rightarrow f\)  is any 1-cell, then \(g = t_0g^{\prime }\) , for some \(g^{\prime }:2 \rightarrow 0\) , which is necessarily of one of the following three types.

\(g^{\prime } \cong d_0d_0\)  via unique isomorphism

\(g^{\prime } \cong d_0d_2\)  via unique isomorphism

\(g^{\prime } \cong d_1d_2\)  via unique isomorphism

It suffices therefore to find the necessary and sufficient conditions such that for a choice of \(g_1^{\prime },g_2^{\prime }\)  among the three types above, there is a unique isomorphism \(t_0g_1^{\prime } \cong t_0g_2^{\prime }\) . There are three types of pairs of 1-cells to consider.

In the case where \(g_1^{\prime } \cong g_2^{\prime }\cong d_0d_0\) , observe that the diagram of unique isomorphisms between 1-cells
t0d0d0 [r, "* d0", ""'] [d, "t0*0"', ""] t0d1d0 [r, "t0*(1)-1", ""']  t0d0d2
t0d0d1 [r, "* d1"', ""]
 t0d1d1 [r, "t0*2"', ""]
 t0d1d2, [u, "-1* d2"', ""]
.
commutes by uniqueness of isomorphisms between 1-cells. It follows that there is a unique isomorphism \(t_0g_1^{\prime } \cong t_0g_2^{\prime }\)  if and only if
\((t_0 (\alpha _1)^{-1}) \circ (\theta * d_0) = (\theta ^{-1} * d_2) \circ (t_0 * \alpha _2) \circ (\theta * d_1) \circ (t_0 * \alpha _0)\) 

When \(g_1^{\prime } \cong d_0d_0, g_2^{\prime }\cong d_0d_2\) , the same commuting diagram as in the previous case implies that the same necessary and sufficient conditions apply in this case.

When \(g_1^{\prime }=d_0d_2\) , \(g_2^{\prime }=d_1d_2\) , the same commuting diagram again leads to the same necessary and sufficient conditions.

It follows that \(Q\Big (\big ((\Delta _{\le 2})^{op}\big )^{\rhd }\Big )(i,f)\)  is contractible for \(i = 0,1,2\)  if and only if
\(\theta * s_0&= t_0*((\beta _0)^{-1} \circ \beta _1)\\(t_0 (\alpha _1)^{-1}) \circ (\theta * d_0)&= (\theta ^{-1} * d_2) \circ (t_0 * \alpha _2) \circ (\theta * d_1) \circ (t_0 * \alpha _0),\) 

which finishes the proof.
Remark 1.19 The results of this section (and in particular lemma:trunc simpl lift) focused on the case where \({B}\)  is a 2-category, as we apply these results only to the 2-category of categories (in thm:main theorem). However, following [40]}, an
analogous result should certainly hold for bicategories and pseudofunctors of bicategories (called homomorphisms of bicategories in [40]}).

